3.1 Introduction

This chapter specifies the "presentation" elements of
MathML, which can be used to describe the layout structure of mathematical
notation.

3.1.1 What Presentation Elements Represent

Presentation elements correspond to the "constructors"
of traditional mathematical notation - that is, to the basic
kinds of symbols and expression-building structures out of which any
particular piece of traditional mathematical notation is built.
Because of the importance of traditional visual notation, the
descriptions of the notational constructs the elements represent are
usually given here in visual terms. However, the elements are
medium-independent in the sense that they have been designed to
contain enough information for good spoken renderings as well. Some
attributes of these elements may make sense only for visual media, but
most attributes can be treated in an analogous way in audio as well
(for example, by a correspondence between time duration and horizontal
extent).

MathML presentation elements only suggest (i.e. do not require)
specific ways of rendering in order to allow for medium-dependent
rendering and for individual preferences of style. This specification
describes suggested visual rendering rules in some detail, but a
particular MathML renderer is free to use its own rules as long as its
renderings are intelligible.

The presentation elements are meant to express the syntactic
structure of mathematical notation in much the same way as titles, sections,
and paragraphs capture the higher-level syntactic structure of a
textual document. Because of this, for example, a single row of
identifiers and operators, such as "x + a /
b", will often be represented not just by one
mrow element (which renders as a horizontal row
of its arguments), but by multiple nested mrow
elements corresponding to the nested sub-expressions of which one
mathematical expression is composed - in this case,

Similarly, superscripts are attached not just to the preceding
character, but to the full expression constituting their base. This
structure allows for better-quality rendering of mathematics, especially when
details of the rendering environment such as display widths are not
known to the document author; it also greatly eases automatic
interpretation of the mathematical structures being represented.

Certain MathML characters are used
to name operators or identifiers that in traditional notation render the
same as other symbols, such as &DifferentialD;, &ExponentialE;, or &ImaginaryI;, or
operators that usually render invisibly, such as &InvisibleTimes;, &ApplyFunction;, or
&InvisibleComma;. These are distinct notational
symbols or objects, as evidenced by their distinct spoken renderings and in
some cases by their effects on linebreaking and spacing in visual
rendering, and as such should be represented by the appropriate specific
entity references. For example, the expression represented visually as
"f(x)" would usually be spoken in English as
"f of x" rather than just
"fx"; this is expressible in MathML by
the use of the &ApplyFunction; operator after the
"f", which (in this case) can be aurally rendered as
"of".

The remainder of this section introduces MathML-specific
terminology and conventions used in this chapter.

3.1.2.1 Types of presentation elements

The presentation elements are divided into two classes.
Token elements represent individual symbols, names,
numbers, labels, etc. In general, tokens can have only
characters as content. The
only exceptions are the vertical alignment element malignmark, mglyph,
and entity references.
Layout schemata build expressions out of parts, and can have
only elements as content (except for whitespace, which they ignore). There
are also a few empty elements used only in conjunction with certain layout
schemata.

All individual "symbols" in a mathematical expression should be
represented by MathML token elements. The primary MathML token element
types are identifiers (e.g. variables or function names), numbers, and
operators (including fences, such as parentheses, and separators, such
as commas). There are also token elements for representing text or
whitespace that has more aesthetic than mathematical significance,
and for representing "string literals" for compatibility with
computer algebra systems. Note that although a token element
represents a single meaningful "symbol" (name, number, label,
mathematical symbol, etc.), such symbols may be comprised of more than
one character. For example sin and 24 are
represented by the single tokens <mi>sin</mi>
and <mn>24</mn> respectively.

In traditional mathematical notation, expressions are recursively
constructed out of smaller expressions, and ultimately out of single
symbols, with the parts grouped and positioned using one of a small
set of notational structures, which can be thought of as "expression
constructors". In MathML, expressions are constructed in the same way,
with the layout schemata playing the role of the expression
constructors. The layout schemata specify the way in which
sub-expressions are built into larger expressions. The terminology
derives from the fact that each layout schema corresponds to a
different way of "laying out" its sub-expressions to form a larger
expression in traditional mathematical typesetting.

3.1.2.2 Terminology for other classes of elements and their relationships

A MathML expression is a single instance of any of the
presentation elements with the exception of the empty elements none or mprescripts, or is
a single instance of any of the content elements which are allowed as
content of presentation elements (described in Section 5.2.4 Content Markup Contained in
Presentation Markup). A sub-expression of an expression
E is any MathML expression that is part of the content of
E, whether directly or indirectly,
i.e. whether it is a "child" of E or not.

Since layout schemata attach special meaning to the number and/or
positions of their children, a child of a layout schema is also called
an argument of that element. As a consequence of the
above definitions, the content of a layout schema consists exactly of
a sequence of zero or more elements that are its
arguments.

3.1.3 Required Arguments

Many of the elements described herein require a specific number of
arguments (always 1, 2, or 3). In the detailed descriptions of
element syntax given below, the number of required arguments is
implicitly indicated by giving names for the arguments at various
positions. A few elements have additional requirements on the number
or type of arguments, which are described with the individual
element. For example, some elements accept sequences of zero or more
arguments - that is, they are allowed to occur with no arguments
at all.

Note that MathML elements encoding rendered space do
count as arguments of the elements in which they appear. See Section 3.2.7 Space (mspace) for a discussion of the proper use of such
space-like elements.

3.1.3.1 Inferred mrows

The elements listed in the following table as requiring 1*
argument (msqrt, mstyle,
merror, menclose, mpadded,
mphantom, mtd,
and math) actually
accept any number of arguments. However, if the number of arguments is 0,
or is more than 1, they treat their contents as a single
inferredmrow formed from all
their arguments. Although the math element is
not a presentation element, it is listed below for completeness.

For example,

<mtd>
</mtd>

is treated as if it were

<mtd>
<mrow>
</mrow>
</mtd>

and

<msqrt>
<mo> - </mo>
<mn> 1 </mn>
</msqrt>

is treated as if it were

<msqrt>
<mrow>
<mo> - </mo>
<mn> 1 </mn>
</mrow>
</msqrt>

This feature allows MathML data not to contain (and its authors to
leave out) many mrow elements that would otherwise be
necessary.

In the descriptions in this chapter of the above-listed elements'
rendering behaviors, their content can be assumed to consist of
exactly one expression, which may be an mrow
element formed from their arguments in this manner. However, their
argument counts are shown in the following table as 1*, since
they are most naturally understood as acting on a single
expression.

3.1.3.2 Table of argument requirements

For convenience, here is a table of each element's argument count
requirements, and the roles of individual arguments when these are
distinguished. An argument count of 1* indicates an inferred mrow as described above.

3.1.4 Elements with Special Behaviors

Certain MathML presentation elements exhibit special behaviors in
certain contexts. Such special behaviors are discussed in the
detailed element descriptions below. However, for convenience, some
of the most important classes of special behavior are listed here.

Certain elements, e.g. msup, are able to
embellish operators that are their first argument. These elements are
listed in Section 3.2.5 Operator, Fence, Separator or Accent
(mo), which precisely defines an "embellished
operator" and explains how this affects the suggested rendering rules
for stretchy operators.

Certain elements treat their arguments as the arguments of an
"inferred mrow" if they are not given
exactly one argument, as explained in Section 3.1.3 Required Arguments.

In MathML 1.x, the mtable element could infer
mtr elements around its arguments, and the
mtr element could infer
mtd elements. In MathML 2.0, mtr and mtd elements must
be explicit. However, for backward compatibility renderers may wish
to continue supporting inferred mtr and mtd elements.

3.1.5 Bidirectional Layout

The term 'bidirectional layout' refers to the fact that
letters from certain scripts, in particular Arabic and
Hebrew, are written from right to left, and that mixing
these with numbers or letters from scripts written left-
to-right results in text runs of two differing directions
within the same line or paragraph.

For ordinary text, Unicode defines a bidirectional algorithm
[Bidi]. This algorithm assumes that the order of
characters in a 'backing store' is in logical order (i.e. in the order
it would be pronounced or typed in), and defines how the characters
get reordered for display based on character properties and other
directives. HTML, CSS, XSL, and SVG adopt this algorithm and
provide ways to control it via markup or styling.

In mathematical expressions, bidirectional layout is more difficult
than it is in text. In part, this is due to the 2-dimensional nature
of mathematical layout, and the fact that spatial relationships are
often used to convey meaning in mathematics notation. Another factor is the
lack of established conventions for bidirectional mathematics layout, since
this is relatively uncommon, even in right-to-left contexts.

For these reasons, MathML 2.0 only adopts a restricted version of
the Unicode Bidirectional algorithm, as described in the remainder of
this section.

3.1.5.1 Bidirectional Layout in Token Elements

For MathML token elements that can contain text (mtext, mo, mi, mn and ms), the implicit part of the Unicode
bidirectional algorithm [Bidi] is applied when its
content is rendered visually (i.e. characters are reordered based on
character properties). The base directionality is left-to-right.

The implicit part of the Unicode bidirectional algorithm
is identical to straightforward left-to-right layout if
there is only one character, or if there are no strong
right-to-left characters (i.e. no characters from the
Arabic, Hebrew, or similar scripts).

Applications are not required to apply the Unicode bidirectional
algorithm if they do not render strong right-to-left characters.

Please note that for the transfinite cardinals represented
by Hebrew characters, the codepoints U+2135-U+2138 (ALEF SYMBOL,
BET SYMBOL, GIMEL SYMBOL, DALET SYMBOL) should be used.
These are strong left-to-right.

3.1.5.2 Bidirectional Layout of Mathematics Formulas

MathML 2.0 does not address right-to-left or bidirectional
layout in mathematics formulas. Only left-to-right layout is
supported. Right-to-left layout of mathematical formulas
may be addressed in a future version of MathML.

3.1.6.5 Enlivening Expressions

3.2 Token Elements

Token elements in presentation markup are broadly intended to
represent the smallest units of mathematical notation which carry
meaning. Tokens are roughly analogous to words in text. However,
because of the precise, symbolic nature of mathematical notation, the
various categories and properties of token elements figure prominently in
MathML markup. By contrast, in textual data, individual words rarely
need to be marked up or styled specially.

Frequently tokens consist of a single character denoting a
mathematical symbol. Other cases, e.g. function names, involve
multi-character tokens. Further, because traditional mathematical
notation makes wide use of symbols distinguished by their
typographical properties (e.g. a Fraktur 'g' for a Lie algebra, or a
bold 'x' for a vector), care must be taken to insure that styling
mechanisms respect typographical properties which carry meaning.
Consequently, characters, tokens, and typographical properties of
symbols are closely related to one another in MathML.

3.2.1 MathML characters in
token elements

Character data in MathML markup is only allowed to occur as part of
the content of token elements. The only exception is whitespace
between elements, which is ignored. Token elements can
contain any sequence of zero or more Unicode characters. In
particular, tokens with empty content are allowed, and should
typically render invisibly, with no width except for the normal extra
spacing for that kind of token element. The exceptions to this are
the empty elements mspace and
mglyph.
The mspace element's width depends upon
its attribute values.
The mglyph element
renders using the character described by its attributes.

While all Unicode character data is valid in token element content, MathML
2.0 distinguishes a special subset of named Unicode 3.2 characters,
called MathML characters in this document.
The complete list of MathML characters is defined in
Chapter 6 Characters, Entities and Fonts. MathML characters can be either represented
directly as Unicode character data, or indirectly via numeric or
character entity references. See Chapter 6 Characters, Entities and Fonts for a
discussion of the advantages and disadvantages of numeric
character references versus
entity references. New mathematics characters that arise, or non-standard
glyphs for existing MathML characters, may be represented by means of
the mglyph element.

Apart from the mglyph element, the malignmark element is the only other element
allowed in the content of tokens. See Section 3.5.5 Alignment Markers
for details.

Token elements (other than mspace and
mglyph) should
be rendered as their content (i.e. in the visual case, as a
closely-spaced horizontal row of standard glyphs for the characters in
their content). Rendering algorithms should also take into account the
mathematics style attributes as described below, and modify surrounding
spacing by rules or attributes specific to each type of token
element.

3.2.1.1 Alphanumeric symbol
characters

A large class of mathematical symbols are single letter identifiers
typically used as variable names in formulas. Different font variants
of a letter are treated as separate symbols. For example, a Fraktur
'g' might denote a Lie algebra, while a Roman 'g' denotes the
corresponding Lie group. These letter-like symbols are traditionally
typeset differently than the same characters appearing in text, using
different spacing and ligature conventions. These characters must
also be treated specially by style mechanisms, since arbitrary style
transformations can change meaning in an expression.

For these reasons, Unicode 3.2 contains
more than nine hundred Math Alphanumeric Symbol characters
corresponding to letter-like symbols. These characters are in the
Secondary Multilingual Plane (SMP). See Chapter 6 Characters, Entities and Fonts for
more information. As valid Unicode data, these characters are
permitted in MathML 2.0, and as tools and fonts for them become widely
available, we anticipate they will be the predominant way of denoting
letter-like symbols.

MathML 2.0 also provides an alternative encoding
for these characters using only Basic Multilingual Plane
(BMP) characters together with markup. MathML 2.0 defines a
correspondence between token elements with certain combinations of BMP
character data and the mathvariant attribute and tokens
containing SMP Math Alphanumeric Symbol characters. Processing
applications that accept SMP characters are required to treat the
corresponding BMP and attribute combinations identically. This is particularly important for applications that
support searching and/or equality testing.

3.2.2 Mathematics style attributes common to token
elements

MathML 2.0 introduces four new mathematics style attributes.
These attributes are valid on all presentation token elements except
mspace and mglyph, and
on no other elements except mstyle. The attributes
are:

The mathematics style attributes define logical classes of token
elements. Each class is intended to correspond to a collection of
typographically-related symbolic tokens that have a meaning within a
given math expression, and therefore need to be visually distinguished
and protected from inadvertent document-wide style changes which might
change their meanings.

When MathML rendering takes place in an environment where CSS is
available, the mathematics style attributes can be viewed as
predefined selectors for CSS style rules. See Section 7.1.6 Using CSS with MathML and Appendix G Sample CSS Style Sheet for MathML for further
discussion and a sample CSS style sheet. When CSS is not available,
it is up to the internal style mechanism of the rendering application
to visually distinguish the different logical classes.

Renderers have complete freedom in
mapping mathematics style attributes to specific rendering properties.
However, in practice, the mathematics style attribute names and values
suggest obvious typographical properties, and renderers should attempt
to respect these natural interpretations as far as possible. For
example, it is reasonable to render a token with the
mathvariant attribute set to "sans-serif" in
Helvetica or Arial. However, rendering the token in a Times Roman
font could be seriously misleading and should be avoided.

It is important to note that only certain combinations of
character data and mathvariant attribute values make sense.
For example, there is no clear cut rendering for a 'fraktur' alpha, or
a 'bold italic' Kanji character. By design, the only cases that have
an unambiguous interpretation are exactly the ones that correspond to
SMP Math Alphanumeric Symbol characters, which are ennumerated in
Section 6.2.3 Mathematical Alphanumeric Symbols
Characters. In all other cases, it is suggested
that renderers ignore the value of the mathvariant
attribute if it is present. Similarly, authors should refrain from
using the mathvariant attribute with characters that do not
have SMP counterparts, since renderings may not be useful or
predictable. In the very rare case that it is necessary to specify a
font variant for other characters or symbols within an equation,
external styling mechanisms such as CSS are generally preferable, or
in the last resort, the deprecated style attributes of MathML 1 could be
used.

Token elements also permit id,
xref, class and
style
attributes for compatibility with style sheet
mechanisms, as described in Section 2.4.5 Attributes Shared by all MathML Elements.
However, some care must be taken when using CSS generally. Using CSS to
produce visual effects that alter the meaning of an equation should be
especially avoided, since MathML is used in many non-CSS environments.
Similarly, care should be taken to insure arbitrary document-wide
style transformations do not affect mathematics expressions in such a way
that meaning is altered.

Since MathML expressions are often embedded in a textual data
format such as XHTML, the surrounding text and the MathML must share
rendering attributes such as font size, so that the renderings will be
compatible in style. For this reason, most attribute values affecting
text rendering are inherited from the rendering environment, as shown
in the "default" column in the table above. (In
cases where the surrounding text and the MathML are being rendered by
separate software, e.g. a browser and a plug-in, it is also important
for the rendering environment to provide the MathML renderer with
additional information, such as the baseline position of surrounding
text, which is not specified by any MathML attributes.)
Note, however, that MathML 2.0 doesn't specify the mechanism by which
style information is inherited from the rendering environment.
For example, one browser plug-in might choose to rely
completely on the CSS inheritance mechanism and use the fully resolved
CSS properties for rendering, while another application might only consult
a style environment at the root node, and then use its own internal style
inheritance rules.

Most MathML renderers will probably want to rely on some degree to
additional, internal style processing algorithms. In particular,
inheritance of the mathvariant attribute does
not follow the CSS model. The default value for this attribute is "normal" (non-slanted) for all tokens except
mi.
For mi tokens, the default depends on the number of
characters in tokens' content. (The deprecatedfontslant attribute also behaves this way.) See
Section 3.2.3 Identifier (mi) for details.

3.2.2.1 Deprecated style attributes on token
elements

The MathML 1.01 style attributes listed below have been deprecated in MathML 2.0.
In rendering environments that support CSS, it is preferable to use
CSS to control the rendering properties corresponding to these
attributes. However as explained above, direct manipulation of these
rendering properties by whatever means should usually be avoided.

If both a new mathematics style attribute and conflicting deprecated
attributes are given, the new math style attribute value should be
used. For example

<mi fontweight='bold' mathvariant='normal'> a </mi>

should render in a normal weight font, and

<mi fontweight='bold' mathvariant='sans-serif'> a </mi>

should render in a normal weight sans serif font. In the example

<mi fontweight='bold' mathvariant='fraktur'> a1 </mi>

the mathvariant attribute still overrides fontweight attribute, even though "fraktur" generally shouldn't be applied to a '1'
since there is no corresponding SMP Math Alphanumeric Symbol
character. In the absence of fonts containing Fraktur digits,
this would probably render as a Fraktur 'a' followed by a Roman '1' in
most renderers.

The new mathematics style attributes also override deprecated 1.01
style attribute values that are inherited. Thus

<mstyle fontstyle='italic'>
<mi mathvariant='bold'> a </mi>
</mstyle>

renders in a bold upright font, not a bold italic font.

At the same time, the MathML 1.01 attributes still serve a
purpose. Since they correspond directly to rendering properties needed
for mathematics layout, they are very useful for describing MathML layout
rules and algorithms. For this reason, and for backward compatibility,
the MathML rendering rules suggested in this chapter continue to be
described in terms of the rendering properties described by these
MathML 1.01 style attributes.

The deprecated attributes are:

Name

values

default

fontsize

number v-unit

inherited

fontweight

normal | bold

inherited

fontstyle

normal | italic

normal (except on<mi>)

fontfamily

string | css-fontfamily

inherited

color

#rgb | #rrggbb | html-color-name

inherited

The fontsize attribute specifies the desired
font size. v-unit represents a unit of
vertical length (see Section 2.4.4.3 CSS-compatible attributes). The most common
unit for specifying font sizes in typesetting is pt
(points).

If the requested size of the current font is not available, the
renderer should approximate it in the manner likely to lead to the
most intelligible, highest quality rendering.

The value of the fontfamily attribute should
be the name of a font that may be available to a MathML renderer, or
information that permits the renderer to select a font in some manner;
acceptable values and their meanings are dependent on the specific
renderer and rendering environment in use, and are not specified by
MathML (but see the note about css-fontfamily
below). (Note that the renderer's mechanism for finding fonts by name
may be case-sensitive.)

If the value of fontfamily is not recognized by a
particular MathML renderer, this should never be interpreted as a
MathML error; rather, the renderer should either use a font that it
considers to be a suitable substitute for the requested font, or
ignore the attribute and act as if no value had been given.

Note that any use of the fontfamily
attribute is unlikely to be portable across all MathML renderers. In
particular, it should never be used to try to achieve the effect of a
reference to a non-ASCII MathML character (for example, by using a
reference to a character in some symbol font that maps ordinary
characters to glyphs for non-ASCII characters). As a corollary to this
principle, MathML renderers should attempt to always produce
intelligible renderings for the MathML characters listed in Chapter 6 Characters, Entities and Fonts, even when these characters are not available in the
font family indicated. Such a rendering is always possible - as
a last resort, a character can be rendered to appear as an XML-style
entity reference using one of the entity names given for the same
character in Chapter 6 Characters, Entities and Fonts.

The symbol css-fontfamily refers to a legal
value for the font-family property in CSS,
which is a comma-separated list of alternative font family names or
generic font types in order of preference, as documented in more
detail in CSS[CSS2].
MathML renderers are encouraged to make use of the CSS
syntax for specifying fonts when this is practical in their rendering
environment, even if they do not otherwise support CSS. (See also the
subsection CSS-compatible attributes within Section 2.4.4.3 CSS-compatible attributes).

3.2.2.2 Color-related attributes

The mathcolor (and deprecated color) attribute controls the color in which the
content of tokens is rendered. Additionally, when inherited from
mstyle or from a MathML expression's rendering
environment, it controls the color of all other drawing by MathML
elements, including the lines or radical signs that can be drawn by
mfrac, mtable, or
msqrt.

The values of mathcolor, color,
mathbackground, and background can be specified
as a string consisting of "#" followed without intervening whitespace
by either 1-digit or 2-digit hexadecimal values for the red, green,
and blue components, respectively, of the desired color. The same number of digits must be used for each
component. No whitespace is allowed between the '#' and the
hexadecimal values. The hexadecimal digits are not
case-sensitive. The possible 1-digit values range from 0 (component
not present) to F (component fully present), and the possible 2-digit
values range from 00 (component not present) to FF (component fully
present), with the 1-digit value x being equivalent to the
2-digit value xx (rather than x0).

These attributes can also be specified as an
html-color-name, which is defined below. Additionally, the keyword "transparent" may
be used for the background attribute.

The color syntax described above is a subset of the syntax of the color and background-color
properties of CSS. The background-color syntax
is in turn a subset of the full CSS background
property syntax, which also permits specification of (for example)
background images with optional repeats. The more general attribute name
background is used in MathML to facilitate possible
extensions to the attribute's scope in future versions of MathML.

Color values on either attribute can also be specified as an html-color-name, that is, as one of the color-name
keywords defined in [HTML4]
("aqua",
"black",
"blue",
"fuchsia",
"gray",
"green",
"lime",
"maroon",
"navy",
"olive",
"purple",
"red",
"silver",
"teal",
"white", and
"yellow").
Note that the color name keywords are not case-sensitive, unlike most
keywords in MathML attribute values for compatibility with CSS and HTML.

The suggested MathML visual rendering rules do not define the
precise extent of the region whose background is affected by using the
background attribute on mstyle,
except that, when mstyle's content does not have
negative dimensions and its drawing region is not overlapped by other
drawing due to surrounding negative spacing, this region should lie
behind all the drawing done to render the content of the
mstyle, but should not lie behind any of the
drawing done to render surrounding expressions. The effect of overlap
of drawing regions caused by negative spacing on the extent of the
region affected by the background attribute is not
defined by these rules.

3.2.3 Identifier (mi)

3.2.3.1 Description

An mi element represents a symbolic name or
arbitrary text that should be rendered as an identifier. Identifiers
can include variables, function names, and symbolic constants.

Not all "mathematical identifiers" are represented by
mi elements - for example, subscripted or primed
variables should be represented using msub or
msup respectively. Conversely, arbitrary text
playing the role of a "term" (such as an ellipsis in a summed series)
can be represented using an mi element, as shown
in an example in Section 3.2.6.4 Mixing text and mathematics.

It should be stressed that mi is a
presentation element, and as such, it only indicates that its content
should be rendered as an identifier. In the majority of cases, the
contents of an mi will actually represent a
mathematical identifier such as a variable or function name. However,
as the preceding paragraph indicates, the correspondence between
notations that should render like identifiers and notations that are
actually intended to represent mathematical identifiers is not
perfect. For an element whose semantics is guaranteed to be that of an
identifier, see the description of ci in
Chapter 4 Content Markup.

A typical graphical renderer would render an mi element as the characters in its content, with
no extra spacing around the characters (except spacing associated with
neighboring elements). The default mathvariant
and fontstyle would (typically) be "normal" (non-slanted) unless the content is a single
character, in which case it would be "italic". Note that this rule for mathvariant and fontstyle
attributes is specific to mi elements; the
default value for the mathvariant and fontstyle attributes on other MathML token elements
is "normal".

3.2.3.3 Examples

An mi element with no content is allowed;
<mi></mi> might, for example, be used by an
"expression editor" to represent a location in a MathML expression
which requires a "term" (according to conventional syntax for
mathematics) but does not yet contain one.

Identifiers include function names such as
"sin". Expressions such as "sin x"
should be written using the &ApplyFunction; operator
(which also has the short name &af;) as shown below;
see also the discussion of invisible operators in Section 3.2.5 Operator, Fence, Separator or Accent
(mo).

<mrow>
<mi> sin </mi>
<mo> &ApplyFunction; </mo>
<mi> x </mi>
</mrow>

Miscellaneous text that should be treated as a "term" can also be
represented by an mi element, as in:

When an mi is used in such exceptional
situations, explicitly setting the fontstyle attribute
may give better results than the default behavior of some
renderers.

The names of symbolic constants should be represented as
mi elements:

<mi> &pi; </mi>
<mi> &ImaginaryI; </mi>
<mi> &ExponentialE; </mi>

Use of special entity references for such constants can simplify
the interpretation of MathML presentation elements.
See Chapter 6 Characters, Entities and Fonts for a complete list of character entity
references in MathML.

3.2.4 Number (mn)

3.2.4.1 Description

An mn element represents a "numeric
literal" or other data that should be rendered as a numeric
literal. Generally speaking, a numeric literal is a sequence of digits,
perhaps including a decimal point, representing an unsigned integer or real
number.

The mathematical concept of a "number" can be quite
subtle and involved, depending on the context. As a consequence, not all
mathematical numbers should be represented using mn; examples of mathematical numbers that should be
represented differently are shown below, and include
complex numbers, ratios of numbers shown as fractions, and names of numeric
constants.

Conversely, since mn is a presentation
element, there are a few situations where it may desirable to include
arbitrary text in the content of an mn that
should merely render as a numeric literal, even though that content
may not be unambiguously interpretable as a number according to any
particular standard encoding of numbers as character sequences. As a
general rule, however, the mn element should be
reserved for situations where its content is actually intended to
represent a numeric quantity in some fashion. For an element whose
semantics are guaranteed to be that of a particular kind of
mathematical number, see the description of cn in
Chapter 4 Content Markup.

3.2.4.2 Attributes

A typical graphical renderer would render an
mn element as the characters of its content, with
no extra spacing around them (except spacing from neighboring elements
such as mo). Unlike mi,
mn elements are (typically) rendered in an
unslanted font by default, regardless of their content.

3.2.4.3 Examples

3.2.4.4 Numbers that should not be written
using mn alone

Many mathematical numbers should be represented using presentation
elements other than mn alone; this includes
complex numbers, ratios of numbers shown as fractions, and
names of numeric constants. Examples of MathML representations of
such numbers include:

3.2.5 Operator, Fence, Separator or Accent
(mo)

3.2.5.1 Description

An mo element represents an operator or
anything that should be rendered as an operator. In general, the
notational conventions for mathematical operators are quite
complicated, and therefore MathML provides a relatively sophisticated
mechanism for specifying the rendering behavior of an
mo element. As a consequence, in MathML the list
of things that should "render as an operator" includes a number of
notations that are not mathematical operators in the ordinary
sense. Besides ordinary operators with infix, prefix, or postfix
forms, these include fence characters such as braces, parentheses, and
"absolute value" bars, separators such as comma and semicolon, and
mathematical accents such as a bar or tilde over a symbol.

The term "operator" as used in the present chapter means
any symbol or notation that should render as an operator, and that is
therefore representable by an mo element. That is,
the term "operator" includes any ordinary operator, fence,
separator, or accent unless otherwise specified or clear from the
context.

All such symbols are represented in MathML with mo elements since they are subject to essentially the
same rendering attributes and rules; subtle distinctions in the rendering
of these classes of symbols, when they exist, are supported using the
boolean attributes fence, separator and accent, which can be
used to distinguish these cases.

A key feature of the mo element is that its
default attribute values are set on a case-by-case basis from an
"operator dictionary" as explained below. In particular, default
values for fence, separator and
accent can usually be found in the operator dictionary
and therefore need not be specified on each mo
element.

Note that some mathematical operators are represented not by mo elements alone, but by mo
elements "embellished" with (for example) surrounding
superscripts; this is further described below. Conversely, as presentation
elements, mo elements can contain arbitrary text,
even when that text has no standard interpretation as an operator; for an
example, see the discussion "Mixing text and mathematics" in
Section 3.2.6 Text (mtext). See also Chapter 4 Content Markup for
definitions of MathML content elements that are guaranteed to have the
semantics of specific mathematical operators.

set by position of operator in an mrow (rule given below);
used with mo content to index operator dictionary

fence

true | false

set by dictionary (false)

separator

true | false

set by dictionary (false)

lspace

number h-unit | namedspace

set by dictionary (thickmathspace)

rspace

number h-unit | namedspace

set by dictionary (thickmathspace)

stretchy

true | false

set by dictionary (false)

symmetric

true | false

set by dictionary (true)

maxsize

number [ v-unit | h-unit ] | namedspace | infinity

set by dictionary (infinity)

minsize

number [ v-unit | h-unit ] | namedspace

set by dictionary (1)

largeop

true | false

set by dictionary (false)

movablelimits

true | false

set by dictionary (false)

accent

true | false

set by dictionary (false)

h-unit represents a unit of horizontal
length, and v-unit represents a unit of vertical
length (see
Section 2.4.4.2 Attributes with units).
namedspace is one of
"veryverythinmathspace",
"verythinmathspace",
"thinmathspace",
"mediummathspace",
"thickmathspace",
"verythickmathspace", or
"veryverythickmathspace".
These values can be set by using the mstyle element
as is further discussed in Section 3.3.4 Style Change (mstyle).

If no unit is given with maxsize or minsize, the number is a multiplier of the normal size
of the operator in the direction (or directions) in which it stretches.
These attributes are further explained below.

Typical graphical renderers show all mo
elements as the characters of their content, with additional spacing
around the element determined from the attributes listed
above. Detailed rules for determining operator spacing in visual
renderings are described in a subsection below. As always, MathML does
not require a specific rendering, and these rules are provided as
suggestions for the convenience of implementors.

Renderers without access to complete fonts for the MathML character
set may choose not to render an mo element as
precisely the characters in its content in some cases. For example,
<mo> &le; </mo> might be rendered as
<= to a terminal. However, as a general rule,
renderers should attempt to render the content of an
mo element as literally as possible.
That is,
<mo> &le; </mo> and
<mo> &lt;= </mo> should render differently.
(The first one should render as a single character
representing a less-than-or-equal-to sign, and the second one as the
two-character sequence <=.)

3.2.5.3 Examples with ordinary operators

3.2.5.4 Examples with fences and separators

Note that the mo elements in these examples
don't need explicit fence or separator attributes, since these can be found using the
operator dictionary as described below. Some of these examples could also
be encoded using the mfenced element described in
Section 3.3.8 Expression Inside Pair of Fences
(mfenced).

3.2.5.5 Invisible operators

Certain operators that are "invisible" in traditional
mathematical notation should be represented using specific entity
references within mo elements, rather than simply
by nothing. The entity references used for these "invisible
operators" are:

The reasons for using specific mo elements for
invisible operators include:

such operators should often have specific effects on visual
rendering (particularly spacing and linebreaking rules) that are not
the same as either the lack of any operator, or spacing represented by
mspace or mtext
elements;

these operators should often have specific audio renderings
different than that of the lack of any operator;

automatic semantic interpretation of MathML presentation elements
is made easier by the explicit specification of such operators.

For example, an audio renderer might render f(x)
(represented as in the above examples) by speaking "f of x", but use
the word "times" in its rendering of xy.
Although its rendering must still be different depending on the structure
of neighboring elements (sometimes leaving out "of" or
"times" entirely), its task is made much easier by the use of
a different mo element for each invisible
operator.

3.2.5.6 Names for other special operators

MathML also includes &DifferentialD; for use
in an mo element representing the differential
operator symbol usually denoted by "d". The reasons for
explicitly using this special entity are similar to those for using
the special entities for invisible operators described in the
preceding section.

3.2.5.7 Detailed rendering rules for mo elements

Typical visual rendering behaviors for mo
elements are more complex than for the other MathML token elements, so
the rules for rendering them are described in this separate
subsection.

Note that, like all rendering rules in MathML, these rules are
suggestions rather than requirements. Furthermore, no attempt is made
to specify the rendering completely; rather, enough information is
given to make the intended effect of the various rendering attributes
as clear as possible.

3.2.5.7.1 The operator dictionary

Many mathematical symbols, such as an integral sign, a plus sign,
or a parenthesis, have a well-established, predictable, traditional
notational usage. Typically, this usage amounts to certain default
attribute values for mo elements with specific
contents and a specific form attribute. Since these
defaults vary from symbol to symbol, MathML anticipates that renderers
will have an "operator dictionary" of default attributes for
mo elements (see Appendix F Operator Dictionary) indexed by each
mo element's content and form
attribute. If an mo element is not listed in the
dictionary, the default values shown in parentheses in the table of
attributes for mo should be used, since these
values are typically acceptable for a generic operator.

Some operators are "overloaded", in the sense that they can occur
in more than one form (prefix, infix, or postfix), with possibly
different rendering properties for each form. For example, "+" can be
either a prefix or an infix operator. Typically, a visual renderer
would add space around both sides of an infix operator, while only on
the left of a prefix operator. The form attribute allows
specification of which form to use, in case more than one form is
possible according to the operator dictionary and the default value
described below is not suitable.

3.2.5.7.2 Default value of the
form attribute

The form attribute does not usually have to be
specified explicitly, since there are effective heuristic rules for
inferring the value of the form attribute from the
context. If it is not specified, and there is more than one possible
form in the dictionary for an mo element with
given content, the renderer should choose which form to use as follows
(but see the exception for embellished operators, described later):

If the operator is the first argument in an mrow of length (i.e. number of arguments) greater than
one (ignoring all space-like arguments (see Section 3.2.7 Space (mspace)) in the
determination of both the length and the first argument), the prefix form
is used;

if it is the last argument in an mrow of
length greater than one (ignoring all space-like arguments), the postfix
form is used;

in all other cases, including when the operator is not part of an
mrow, the infix form is used.

Note that these rules make reference to the
mrow in which the mo
element lies. In some situations, this mrow
might be an inferred mrow implicitly present
around the arguments of an element such as
msqrt or mtd.

Opening (left) fences should have form="prefix",
and closing (right) fences should have form="postfix";
separators are usually "infix", but not always,
depending on their surroundings. As with ordinary operators,
these values do not usually need to be specified explicitly.

If the operator does not occur in the dictionary with the specified
form, the renderer should use one of the forms that is available
there, in the order of preference: infix, postfix, prefix; if no forms
are available for the given mo element content, the
renderer should use the defaults given in parentheses in the table of
attributes for mo.

3.2.5.7.3 Exception for embellished operators

There is one exception to the above rules for choosing an mo element's default form
attribute. An mo element that is
"embellished" by one or more nested subscripts, superscripts,
surrounding text or whitespace, or style changes behaves differently. It is
the embellished operator as a whole (this is defined precisely, below)
whose position in an mrow is examined by the above
rules and whose surrounding spacing is affected by its form, not the mo element at its core; however, the attributes
influencing this surrounding spacing are taken from the mo element at the core (or from that element's
dictionary entry).

For example, the "+4" in
a+4b
should be considered an infix operator as a whole, due to its position
in the middle of an mrow, but its rendering
attributes should be taken from the mo element
representing the "+", or when those are not specified explicitly,
from the operator dictionary entry for <mo form="infix"> +
</mo>.
The precise definition of an "embellished operator" is:

an mo element;

or one of the elements
msub,
msup,
msubsup,
munder,
mover,
munderover,
mmultiscripts,
mfrac, or
semantics
(Section 4.2.6 Syntax and Semantics), whose first argument exists and is an embellished
operator;

or one of the elements
mstyle,
mphantom, or
mpadded,
such that an mrow containing the same
arguments would be an embellished operator;

or an maction element whose selected
sub-expression exists and is an embellished operator;

or an mrow whose arguments consist (in any order)
of one embellished operator and zero or more space-like elements.

Note that this definition permits nested embellishment only when
there are no intervening enclosing elements not in the above list.

The above rules for choosing operator forms and defining
embellished operators are chosen so that in all ordinary cases it will
not be necessary for the author to specify a form
attribute.

3.2.5.7.4 Rationale for definition of embellished operators

The following notes are included as a rationale for certain aspects
of the above definitions, but should not be important for most users
of MathML.

An mfrac is included as an
"embellisher" because of the common notation for a
differential operator:

Since the definition of embellished operator affects the use of the
attributes related to stretching, it is important that it includes
embellished fences as well as ordinary operators; thus it applies to
any mo element.

Note that an mrow containing a single argument
is an embellished operator if and only if its argument is an embellished
operator. This is because an mrow with a single
argument must be equivalent in all respects to that argument alone (as
discussed in Section 3.3.1 Horizontally Group Sub-Expressions
(mrow)). This means that an mo element that is the sole argument of an mrow will determine its default form attribute based on that mrow's position in a surrounding, perhaps inferred, mrow (if there is one), rather than based on its own
position in the mrow in which it is the sole
argument.

Note that the above definition defines every
mo element to be "embellished" - that is,
"embellished operator" can be considered (and implemented in
renderers) as a special class of MathML expressions, of which
mo is a specific case.

3.2.5.7.5 Spacing around an operator

The amount of space added around an operator (or embellished operator),
when it occurs in an mrow, can be directly
specified by the lspace and rspace
attributes. By convention, operators that tend to bind tightly to their
arguments have smaller values for spacing than operators that tend to bind
less tightly. This convention should be followed in the operator dictionary
included with a MathML renderer. In TEX, these values can only be one of
three values; typically they are 3/18em, 4/18em, and 5/18em. MathML does
not impose this limit.

Some renderers may choose to use no space around most operators
appearing within subscripts or superscripts, as is done in TEX.

Non-graphical renderers should treat spacing attributes, and other
rendering attributes described here, in analogous ways for their
rendering medium. For example, more space might translate into a
longer pause in an audio rendering.

3.2.5.8 Stretching of operators, fences and accents

Four attributes govern whether and how an operator (perhaps embellished)
stretches so that it matches the size of other elements: stretchy, symmetric, maxsize, and minsize. If an
operator has the attribute stretchy="true", then it (that is, each character in its content)
obeys the stretching rules listed below, given the constraints imposed by
the fonts and font rendering system. In practice, typical renderers will
only be able to stretch a small set of characters, and quite possibly will
only be able to generate a discrete set of character sizes.

There is no provision in MathML for specifying in which direction
(horizontal or vertical) to stretch a specific character or operator;
rather, when stretchy="true" it
should be stretched in each direction for which stretching is possible. It
is up to the renderer to know in which directions it is able to stretch
each character. (Most characters can be stretched in at most one direction
by typical renderers, but some renderers may be able to stretch certain
characters, such as diagonal arrows, in both directions independently.)

The minsize and maxsize
attributes limit the amount of stretching (in either direction). These two
attributes are given as multipliers of the operator's normal size in the
direction or directions of stretching, or as absolute sizes using units.
For example, if a character has maxsize="3", then it
can grow to be no more than three times its normal (unstretched) size.

The symmetric attribute governs whether the
height and
depth above and below the axis of the
character are forced to be equal
(by forcing both height and depth to become the maximum of the two).
An example of a situation where one might set
symmetric="false"
arises with parentheses around a matrix not aligned on the axis, which
frequently occurs when multiplying non-square matrices. In this case, one
wants the parentheses to stretch to cover the matrix, whereas stretching
the parentheses symmetrically would cause them to protrude beyond one edge
of the matrix. The symmetric attribute only applies
to characters that stretch vertically (otherwise it is ignored).

If a stretchy mo element is embellished (as defined
earlier in this section), the mo element at its core is
stretched to a size based on the context of the embellished operator
as a whole, i.e. to the same size as if the embellishments were not
present. For example, the parentheses in the following example (which
would typically be set to be stretchy by the operator dictionary) will be
stretched to the same size as each other, and the same size they would
have if they were not underlined and overlined, and furthermore will
cover the same vertical interval:

Note that each parenthesis is sized independently; if only one of
them had maxsize="1", they would render with different
sizes.

3.2.5.8.2 Vertical Stretching Rules

If a stretchy operator is a direct sub-expression of an mrow element, or is the sole direct sub-expression of an
mtd element in some row of a table, then it should
stretch to cover the height and depth (above and below the axis) of the non-stretchy direct sub-expressions in the
mrow element or table row, unless stretching is
constrained by minsize or maxsize attributes.

In the case of an embellished stretchy operator, the preceding
rule applies to the stretchy operator at its core.

If symmetric="true",
then the maximum of the height and depth is used to determine the size,
before application of the minsize or maxsize attributes.

The preceding rules also apply in situations where the mrow element is inferred.

Most common opening and closing fences are defined in the operator
dictionary to stretch by default; and they stretch vertically. Also,
operators such as &sum;, &int;,
/, and vertical arrows stretch vertically by default.

In the case of a stretchy operator in a table cell (i.e. within an
mtd element), the above rules assume each cell of
the table row containing the stretchy operator covers exactly one row.
(Equivalently, the value of the rowspan attribute is
assumed to be 1 for all the table cells in the table row, including
the cell containing the operator.) When this is not the case, the
operator should only be stretched vertically to cover those table
cells that are entirely within the set of table rows that the
operator's cell covers. Table cells that extend into rows not covered
by the stretchy operator's table cell should be ignored. See
Section 3.5.4.2 Attributes for details about the rowspan attribute.

3.2.5.8.3 Horizontal Stretching Rules

If a stretchy operator, or an embellished stretchy operator,
is a direct sub-expression of an munder,
mover, or munderover element,
or if it is the sole direct sub-expression of an mtd element in some
column of a table (see mtable), then it, or the mo element at its core, should stretch to cover
the width of the other direct sub-expressions in the given element (or
in the same table column), given the constraints mentioned above.

If a stretchy operator is a direct sub-expression of an
munder, mover, or
munderover element, or if it is the sole direct
sub-expression of an mtd element in some column of a
table, then it should stretch to cover the width of the other direct
sub-expressions in the given element (or in the same table column),
given the constraints mentioned above.

In the case of an embellished stretchy operator, the preceding
rule applies to the stretchy operator at its core.

By default, most horizontal arrows and some accents stretch
horizontally.

In the case of a stretchy operator in a table cell (i.e. within an
mtd element), the above rules assume each cell of
the table column containing the stretchy operator covers exactly one
column. (Equivalently, the value of the columnspan
attribute is assumed to be 1 for all the table cells in the table row,
including the cell containing the operator.) When this is not the
case, the operator should only be stretched horizontally to cover
those table cells that are entirely within the set of table columns
that the operator's cell covers. Table cells that extend into columns
not covered by the stretchy operator's table cell should be
ignored. See Section 3.5.4.2 Attributes for details about the rowspan attribute.

The rules for horizontal stretching include mtd
elements to allow arrows to stretch for use in commutative diagrams
laid out using mtable. The rules for the horizontal
stretchiness include scripts to make examples such as the following
work:

3.2.5.8.4 Rules Common to both Vertical and Horizontal Stretching

If a stretchy operator is not required to stretch (i.e. if it is
not in one of the locations mentioned above, or if there are no other
expressions whose size it should stretch to match), then it has the
standard (unstretched) size determined by the font and current
fontsize.

If a stretchy operator is required to stretch, but all other expressions
in the containing element (as described above) are also stretchy,
all elements that can stretch should grow to the maximum of the normal
unstretched sizes of all elements in the containing object, if they can
grow that large. If the value of minsize or maxsize prevents this then that (min or max) size is
used.

For example, in an mrow containing nothing but
vertically stretchy operators, each of the operators should stretch to
the maximum of all of their normal unstretched sizes, provided no
other attributes are set that override this behavior. Of course,
limitations in fonts or font rendering may result in the final,
stretched sizes being only approximately the same.

3.2.5.9 Other attributes of mo

The largeop attribute specifies whether the
operator should be drawn larger than normal if displaystyle="true" in the current
rendering environment. This roughly corresponds to TEX's
\displaystyle style setting. MathML uses two attributes, displaystyle and scriptlevel, to
control orthogonal presentation features that TEX encodes into one
"style" attribute with values \displaystyle,
\textstyle, \scriptstyle, and
\scriptscriptstyle. These attributes are discussed further in
Section 3.3.4 Style Change (mstyle) describing the mstyle element.
Note that these attributes can be specified directly on an mstyle element's start tag, but not on most other
elements. Examples of large operators include &int;
and &prod;.

The movablelimits attribute specifies whether
underscripts and overscripts attached to this mo
element should be drawn as subscripts and superscripts when displaystyle="false". movablelimits="false" means that
underscripts and overscripts should never be drawn as subscripts and
superscripts. In general, displaystyle is "true" for displayed mathematics and "false" for inline mathematics. Also, displaystyle is "false" by default
within tables, scripts and fractions, and a few other exceptional
situations detailed in Section 3.3.4 Style Change (mstyle). Thus, operators with
movablelimits="true" will
display with limits (i.e. underscripts and overscripts) in displayed
mathematics, and with subscripts and superscripts in inline mathematics,
tables, scripts and so on. Examples of operators that typically have
movablelimits="true" are &sum;,
&prod;, and lim.

The separator attribute may affect automatic
linebreaking in renderers that position ordinary infix operators at
the beginnings of broken lines rather than at the ends (that is, which
avoid linebreaking just after such operators), since linebreaking
should be avoided just before separators, but is acceptable just after
them.

The fence attribute has no effect in the suggested
visual rendering rules given here; it is not needed for properly
rendering traditional notation using these rules. It is provided so
that specific MathML renderers, especially non-visual renderers, have
the option of using this information.

3.2.6 Text (mtext)

3.2.6.1 Description

An mtext element is used to represent
arbitrary text that should be rendered as itself. In general, the
mtext element is intended to denote commentary
text.

Note that some text with a clearly defined notational role might be
more appropriately marked up using mi or
mo; this is discussed further below.

An mtext element can be used to contain
"renderable whitespace", i.e. invisible characters that are
intended to alter the positioning of surrounding elements. In non-graphical
media, such characters are intended to have an analogous effect, such as
introducing positive or negative time delays or affecting rhythm in an
audio renderer. This is not related to any whitespace in the source MathML
consisting of blanks, newlines, tabs, or carriage returns; whitespace
present directly in the source is trimmed and collapsed, as described in
Section 2.4.6 Collapsing Whitespace in Input. Whitespace that is intended to be rendered
as part of an element's content must be represented by entity references
or mspace elements
(unless it consists only of single blanks between non-whitespace
characters).

3.2.6.3 Examples

3.2.6.4 Mixing text and mathematics

In some cases, text embedded in mathematics could be more appropriately
represented using mo or mi elements.
For example, the expression 'there exists
such that f(x) <1' is equivalent to
and could be represented as:

An example involving an mi element is:
x+x2+···+xn.
In this example, ellipsis should be represented using an mi element, since it takes the place of a term in the
sum; (see Section 3.2.3 Identifier (mi)).

On the other hand, expository text within MathML is best
represented with an mtext element. An example
of this is:

Theorem 1: if x > 1, then
x2 > x.

However, when MathML is
embedded in HTML, or another document markup language, the example is
probably best rendered with only the two inequalities represented as
MathML at all, letting the text be part of the surrounding HTML.

Another factor to consider in deciding how to mark up text is the
effect on rendering. Text enclosed in an mo
element is unlikely to be found in a renderer's operator dictionary,
so it will be rendered with the format and spacing appropriate for an
"unrecognized operator", which may or may not be better than the
format and spacing for "text" obtained by using an
mtext element. An ellipsis entity in an
mi element is apt to be spaced more appropriately
for taking the place of a term within a series than if it appeared in
an mtext element.

3.2.7 Space (mspace)

3.2.7.1 Description

An mspace empty element represents a blank
space of any desired size, as set by its attributes. It can also be
used to make linebreaking suggestions to a visual renderer.
Note that the default values for attributes have been chosen so that
they typically will have no effect on rendering. Thus, the mspace element is generally used with one
or more attribute values explicitly specified.

The linebreak attribute is used to give a
linebreaking hint to a visual renderer. The default value is "auto", which indicates that a renderer should use
whatever default linebreaking algorithm it would normally use. The
meanings of the other values are described in the table below.

Value

Description

newline

start a new line and do not indent

indentingnewline

start a new line and do indent

nobreak

do not allow a linebreak here

goodbreak

if a linebreak is needed on the line, here is a good spot

badbreak

if a linebreak is needed on the line, try to avoid breaking here

In the case when both dimensional attributes and a linebreaking
attribute are set, the linebreaking attribute is ignored.

Note the warning about the legal grouping of "space-like elements"
given below, and the warning about the use of such elements for
"tweaking" or conveying meaning in Section 3.3.6 Adjust Space Around Content
(mpadded). See also the other
elements that can render as whitespace, namely
mtext, mphantom, and
maligngroup.

3.2.7.3 Definition of space-like elements

A number of MathML presentation elements are "space-like" in the
sense that they typically render as whitespace, and do not affect the
mathematical meaning of the expressions in which they appear. As a
consequence, these elements often function in somewhat exceptional
ways in other MathML expressions. For example, space-like elements are
handled specially in the suggested rendering rules for
mo given in Section 3.2.5 Operator, Fence, Separator or Accent
(mo).
The following MathML elements are defined to be "space-like":

an mtext, mspace,
maligngroup, or malignmark
element;

an mstyle, mphantom, or
mpadded element, all of whose direct sub-expressions
are space-like;

an maction element whose selected
sub-expression exists and is space-like;

an mrow all of whose direct
sub-expressions are space-like.

Note that an mphantom is not
automatically defined to be space-like, unless its content is
space-like. This is because operator spacing is affected by whether
adjacent elements are space-like. Since the
mphantom element is primarily intended as an aid
in aligning expressions, operators adjacent to an
mphantom should behave as if they were adjacent
to the contents of the mphantom,
rather than to an equivalently sized area of whitespace.

3.2.7.4 Legal grouping of space-like elements

Authors who insert space-like elements or
mphantom elements into an existing MathML
expression should note that such elements are counted as
arguments, in elements that require a specific number of arguments,
or that interpret different argument positions differently.

Therefore, space-like elements inserted into such a MathML element
should be grouped with a neighboring argument of that element by
introducing an mrow for that purpose. For example,
to allow for vertical alignment on the right edge of the base of a
superscript, the expression

<msup>
<mi> x </mi>
<malignmark edge="right"/>
<mn> 2 </mn>
</msup>

is illegal, because msup must have exactly 2 arguments;
the correct expression would be:

3.2.8 String Literal (ms)

3.2.8.1 Description

The ms element is used to represent
"string literals" in expressions meant to be interpreted by
computer algebra systems or other systems containing "programming
languages". By default, string literals are displayed surrounded by
double quotes. As explained in Section 3.2.6 Text (mtext), ordinary text
embedded in a mathematical expression should be marked up with mtext, or in some cases mo or
mi, but never with ms.

Note that the string literals encoded by ms are made up of characters, mglyphs and
malignmarks rather than "ASCII
strings". For
example, <ms>&amp;</ms> represents a string
literal containing a single character, &, and
<ms>&amp;amp;</ms> represents a string literal
containing 5 characters, the first one of which is
&.

Like all token elements, msdoes trim and
collapse whitespace in its content according to the rules of
Section 2.4.6 Collapsing Whitespace in Input, so whitespace intended to remain in
the content should be encoded as described in that section.

3.2.8.2 Attributes

In visual renderers, the content of an ms
element is typically rendered with no extra spacing added around the
string, and a quote character at the beginning and the end of the
string. By default, the left and right quote characters are both the
standard double quote character &quot;. However,
these characters can be changed with the lquote and
rquote attributes respectively.

The content of ms elements should be rendered with visible
"escaping" of certain characters in the content,
including at least the left and right quoting
characters, and preferably whitespace other than individual
space characters. The intent is for the viewer to see that the
expression is a string literal, and to see exactly which characters
form its content. For example, <ms>double quote is
"</ms> might be rendered as "double quote is \"".

3.2.9 Accessing glyphs for
characters from MathML
(mglyph)

3.2.9.1 Description

Unicode defines a large number of characters used in mathematics,
and in most cases, glyphs representing these characters are widely
available in a variety of fonts. Although these characters should
meet almost all users needs, MathML recognizes that mathematics is not
static and that new characters are added when convenient. Characters
that become well accepted will likely be eventually incorporated by
the Unicode Consortium or other standards bodies, but that is often a
lengthy process. In the meantime, a mechanism is necessary for
accessing glyphs from non-standard fonts representing these characters.

The mglyph element is the means by which users can
directly access glyphs for characters that are not defined by Unicode,
or not known to the renderer. Similarly, the mglyph element
can also be used to select glyph variants for existing Unicode
characters, as might be desirable when a glyph variant has begun to
differentiate itself as a new character by taking on a distinguished
mathematical meaning.

The mglyph element names a specific
glyph, and is valid inside any
MathML leaf content listed in Section 3.1.6 Summary of Presentation Elements (mi, etc.) or
Section 4.2.2 Containers (ci, etc.)
unless otherwise restricted by an attribute (e.g. base=2 to
<cn>). In order
for a visually-oriented renderer to render the character, the renderer
must be told what font to use and what index within that font to
use.

The alt attribute provides an alternate name
for the glyph. If the specified font can't be found, the renderer may
use this name in a warning message or some unknown glyph notation. The
name might also be used by an audio renderer or symbol processing
system and should be chosen to be descriptive. The fontfamily and index
uniquely identify the mglyph; two mglyphs with the same values for fontfamily and index should
be considered identical by applications that must determine whether
two characters/glyphs are identical. The alt
attribute should not be part of the identity test.

The fontfamily and index attributes name a font and position within
that font. All font properties apart from fontfamily are inherited. Variants of the font
(e.g., bold) that may be inherited may be ignored if the variant of
the font is not present. Note that the use of the
fontfamily attribute is deprecated with other token
elements, but not for use with mglyph.

Authors should be aware that rendering requires the fonts
referenced by mglyph, which the MathML
renderer may not have access to or may be not be supported by the
system on which the renderer runs. For these reasons, authors are
encouraged to use mglyph only when
absolutely necessary, and not for stylistic purposes.

3.2.9.3 Example

The following example illustrates how a researcher might use the mglyph construct with an experimental font to work
with braid group notation.

3.3 General Layout Schemata

Besides tokens there are several families of MathML presentation
elements. One family of elements deals with various
"scripting" notations, such as subscript and
superscript. Another family is concerned with matrices and tables. The
remainder of the elements, discussed in this section, describe other basic
notations such as fractions and radicals, or deal with general functions
such as setting style properties and error handling.

3.3.1 Horizontally Group Sub-Expressions
(mrow)

3.3.1.1 Description

An mrow element is used to group together any
number of sub-expressions, usually consisting of one or more mo elements acting as "operators" on one
or more other expressions that are their "operands".

3.3.1.2 Attributes

mrow elements are typically rendered visually
as a horizontal row of their arguments, left to right in the order in
which the arguments occur, or audibly as a sequence of renderings of
the arguments. The description in Section 3.2.5 Operator, Fence, Separator or Accent
(mo) of suggested rendering
rules for mo elements assumes that all horizontal
spacing between operators and their operands is added by the rendering
of mo elements (or, more generally, embellished
operators), not by the rendering of the mrows
they are contained in.

MathML is designed to allow renderers to automatically
linebreak expressions (that is, to break excessively long
expressions into several lines), without requiring authors to specify
explicitly how this should be done. This is because linebreaking
positions can't be chosen well without knowing the width of the
display device and the current font size, which for many uses of
MathML will not be known except by the renderer at the time of each
rendering.

Determining good positions for linebreaks is complex, and rules for
this are not described here; whether and how it is done is up to each
MathML renderer. Typically, linebreaking will involve selection of
"good" points for insertion of linebreaks between successive
arguments of mrow elements.

Although MathML does not require linebreaking or specify a
particular linebreaking algorithm, it has several features designed to
allow such algorithms to produce good results. These include the use
of special entities for certain operators, including invisible
operators (see Section 3.2.5 Operator, Fence, Separator or Accent
(mo)), or for providing hints related to
linebreaking when necessary (see Section 3.2.6 Text (mtext)), and the ability to
use nested mrows to describe sub-expression
structure (see below).

3.3.1.2.1 mrow of one argument

MathML renderers are required to treat an mrow
element containing exactly one argument as equivalent in all ways to
the single argument occurring alone, provided there are no attributes
on the mrow element's start tag. If there are
attributes on the mrow element's start tag, no
requirement of equivalence is imposed. This equivalence condition is
intended to simplify the implementation of MathML-generating software
such as template-based authoring tools. It directly affects the
definitions of embellished operator and space-like element and the
rules for determining the default value of the form
attribute of an mo element;
see Section 3.2.5 Operator, Fence, Separator or Accent
(mo) and Section 3.2.7 Space (mspace). See also the discussion of equivalence of MathML
expressions in Chapter 7 The MathML Interface.

3.3.1.3 Proper grouping of sub-expressions using mrow

Sub-expressions should be grouped by the document author in the same way
as they are grouped in the mathematical interpretation of the expression;
that is, according to the underlying "syntax tree" of the
expression. Specifically, operators and their mathematical arguments should
occur in a single mrow; more than one operator
should occur directly in one mrow only when they
can be considered (in a syntactic sense) to act together on the interleaved
arguments, e.g. for a single parenthesized term and its parentheses, for
chains of relational operators, or for sequences of terms separated by
+ and -. A precise rule is given below.

Proper grouping has several purposes: it improves display by
possibly affecting spacing; it allows for more intelligent
linebreaking and indentation; and it simplifies possible semantic
interpretation of presentation elements by computer algebra systems,
and audio renderers.

Although improper grouping will sometimes result in suboptimal
renderings, and will often make interpretation other than pure visual
rendering difficult or impossible, any grouping of expressions using
mrow is allowed in MathML syntax; that is,
renderers should not assume the rules for proper grouping will be
followed.

3.3.1.3.1 Precise rule for proper grouping

A precise rule for when and how to nest sub-expressions using
mrow is especially desirable when generating
MathML automatically by conversion from other formats for displayed
mathematics, such as TEX, which don't always specify how sub-expressions
nest. When a precise rule for grouping is desired, the following rule
should be used:

Two adjacent operators (i.e. mo elements,
possibly embellished), possibly separated by operands (i.e. anything
other than operators), should occur in the same
mrow only when the left operator has an infix or
prefix form (perhaps inferred), the right operator has an infix or
postfix form, and the operators are listed in the same group of
entries in the operator dictionary provided in Appendix F Operator Dictionary.
In all other cases, nested mrows should be used.

When forming a nested mrow (during generation
of MathML) that includes just one of two successive operators with
the forms mentioned above (which mean that either operator could in
principle act on the intervening operand or operands), it is necessary
to decide which operator acts on those operands directly (or would do
so, if they were present). Ideally, this should be determined from the
original expression; for example, in conversion from an
operator-precedence-based format, it would be the operator with the
higher precedence. If this cannot be determined directly from the
original expression, the operator that occurs later in the suggested
operator dictionary (Appendix F Operator Dictionary) can be assumed to have
a higher precedence for this purpose.

Note that the above rule has no effect on whether any MathML
expression is valid, only on the recommended way of generating MathML
from other formats for displayed mathematics or directly from written
notation.

The linethickness attribute indicates the thickness of
the horizontal "fraction bar", or "rule", typically used to render
fractions. A fraction with linethickness="0" renders
without the bar, and might be used within binomial coefficients. A
linethickness greater than one might be used with nested
fractions. These cases are shown below:

In general, the value of linethickness can be a
number, as a multiplier of the default thickness of the fraction bar
(the default thickness is not specified by MathML), or a number with a
unit of vertical length (see Section 2.4.4.2 Attributes with units), or one of the keywords
medium (same as 1), thin (thinner than 1,
otherwise up to the renderer), or thick (thicker than 1,
otherwise up to the renderer).

The numalign and
denomalign attributes control the horizontal
alignment of the numerator and denominator respectively. Typically,
numerators and denominators are centered, but a very long numerator or
denominator might be displayed on several lines and a left alignment
might be more appropriate for displaying them.

The bevelled attribute determines whether the
fraction is displayed with the numerator above the denominator
separated by a horizontal line or
whether a diagonal line is used to separate a slightly raised
numerator from a slightly lowered denominator. The latter form
corresponds to the attribute value being "true"
and provides for a more compact form for simple numerator and
denominators. An example illustrating the bevelled form is show below:

The mfrac element sets displaystyle to "false", or if it
was already false increments scriptlevel by 1,
within numerator and denominator. These
attributes are inherited by every element from its rendering environment,
but can be set explicitly only on the mstyle and mtable
elements. (See Section 3.3.4 Style Change (mstyle).)

3.3.3 Radicals (msqrt, mroot)

3.3.3.1 Description

These elements construct radicals. The msqrt element is
used for square roots, while the mroot element is used
to draw radicals with indices, e.g. a cube root. The syntax for these
elements is:

<msqrt> base </msqrt>
<mroot> baseindex </mroot>

The mroot element requires exactly 2 arguments.
However, msqrt accepts any number of arguments; if
this number is not 1, its contents are treated as a single "inferred
mrow" containing its arguments, as described in
Section 3.1.3 Required Arguments.

3.3.3.2 Attributes

The mroot element increments scriptlevel by 2, and sets displaystyle to "false", within
index, but leaves both attributes unchanged within
base. The msqrt element leaves both
attributes unchanged within all its arguments. These attributes are
inherited by every element from its rendering environment, but can be set
explicitly only on mstyle. (See Section 3.3.4 Style Change (mstyle).)

3.3.4 Style Change (mstyle)

3.3.4.1 Description

The mstyle element is used to make style
changes that affect the rendering of its
contents. mstyle can be given any attribute
accepted by any MathML presentation element provided that the
attribute value is inherited, computed or has a default value;
presentation element attributes whose values are required are not
accepted by the mstyle element. In addition
mstyle can also be given certain special
attributes listed below.

The mstyle element accepts any number of
arguments. If this number is not 1, its contents are treated as a single
"inferred mrow" formed from all its
arguments, as described in Section 3.1.3 Required Arguments.

Loosely speaking, the effect of the mstyle element
is to change the default value of an attribute for the elements it
contains. Style changes work in one of several ways, depending on
the way in which default values are specified for an attribute.
The cases are:

Some attributes, such as displaystyle or
scriptlevel (explained below), are inherited
from the surrounding context when they are not explicitly set. Specifying
such an attribute on an mstyle element sets the
value that will be inherited by its child elements. Unless a child element
overrides this inherited value, it will pass it on to its children, and
they will pass it to their children, and so on. But if a child element does
override it, either by an explicit attribute setting or automatically (as
is common for scriptlevel), the new (overriding)
value will be passed on to that element's children, and then to their
children, etc, until it is again overridden.

Other attributes, such as linethickness on
mfrac, have default values that are not normally
inherited. That is, if the linethickness attribute
is not set on the start tag of an mfrac element,
it will normally use the default value of "1", even if it was
contained in a larger mfrac element that set this
attribute to a different value. For attributes like this, specifying a
value with an mstyle element has the effect of
changing the default value for all elements within its scope. The net
effect is that setting the attribute value with mstyle propagates the change to all the elements it
contains directly or indirectly, except for the individual elements on
which the value is overridden. Unlike in the case of inherited attributes,
elements that explicitly override this attribute have no effect on this
attribute's value in their children.

Another group of attributes, such as stretchy and form, are
computed from operator dictionary information, position in the
enclosing mrow, and other similar data. For
these attributes, a value specified by an enclosing mstyle overrides the value that would normally be
computed.

Note that attribute values inherited from an
mstyle in any manner affect a given element
in the mstyle's content only if that attribute is
not given a value in that element's start tag. On any element for
which the attribute is set explicitly, the value specified on the
start tag overrides the inherited value. The only exception to this
rule is when the value given on the start tag is documented as
specifying an incremental change to the value inherited from that
element's context or rendering environment.

Note also that the difference between inherited and non-inherited
attributes set by mstyle, explained above, only
matters when the attribute is set on some element within the
mstyle's contents that has children also
setting it. Thus it never matters for attributes, such as
color, which can only be set on token elements (or on
mstyle itself).

There are several exceptional elements, mpadded,
mtable, mtr, mlabeledtr and mtd
that have attributes which cannot be set with mstyle. The
mpadded and mtable elements share attribute names
with the mspace element. The mtable, mtr,
mlabeledtr and mtd all share attribute
names. Similarly, mpadded and mo elements also
share an attribute name. Since the syntax for the values these shared
attributes accept differs between elements, MathML specifies that when
the attributes height, width or depth
are specified on an mstyle element, they apply only to
mspace elements, and not the corresponding attributes of
mpadded or mtable. Similarly, when
rowalign, columnalign or groupalign
are specified on an mstyle element, the apply only to the
mtable element, and not the row and cell elements. Finally,
when lspace is set with mstyle, it applies only to
the mo element and not mpadded.

3.3.4.2 Attributes

As stated above, mstyle accepts all
attributes of all MathML presentation elements which do not have
required values. That is, all attributes which have an explicit
default value or a default value which is inherited or computed are
accepted by the mstyle element.

Additionally, mstyle can be given the following special
attributes that are implicitly inherited by every MathML element as
part of its rendering environment:

Name

values

default

scriptlevel

['+' | '-'] unsigned-integer

inherited

displaystyle

true | false

inherited

scriptsizemultiplier

number

0.71

scriptminsize

number v-unit

8pt

background

#rgb | #rrggbb | transparent | html-color-name

transparent

veryverythinmathspace

number h-unit

0.0555556em

verythinmathspace

number h-unit

0.111111em

thinmathspace

number h-unit

0.166667em

mediummathspace

number h-unit

0.222222em

thickmathspace

number h-unit

0.277778em

verythickmathspace

number h-unit

0.333333em

veryverythickmathspace

number h-unit

0.388889em

3.3.4.2.1 scriptlevel and displaystyle

MathML uses two attributes, displaystyle and
scriptlevel, to control orthogonal presentation features
that TEX encodes into one style attribute with values
\displaystyle, \textstyle, \scriptstyle, and \scriptscriptstyle. The
corresponding values of displaystyle and
scriptlevel for those TEX styles would be "true" and
"0", "false" and
"0", "false" and
"1", and "false" and "2",
respectively.

The main effect of the displaystyle attribute is that
it determines the effect of other attributes such as the
largeop and movablescripts attributes of
mo. The main effect of the
scriptlevel attribute is to control the font
size. Typically, the higher the scriptlevel, the smaller
the font size. (Non-visual renderers can respond to the font size in
an analogous way for their medium.) More sophisticated renderers may
also choose to use these attributes in other ways, such as rendering
expressions with displaystyle="false" in a more
vertically compressed manner.

These attributes are given initial values for the outermost
expression of an instance of MathML based on its rendering
environment. A short list of layout schemata described below modify
these values for some of their sub-expressions. Otherwise, values are
determined by inheritance whenever they are not directly specified on
a given element's start tag.

For an instance of MathML embedded in a textual data format (such
as HTML) in "display" mode, i.e. in place of a paragraph,
displaystyle = "true" and
scriptlevel = "0" for the
outermost expression of the embedded MathML; if the
MathML is embedded in "inline" mode, i.e. in place of a character,
displaystyle = "false" and
scriptlevel = "0" for
the outermost expression. See Chapter 7 The MathML Interface for further
discussion of the distinction between "display" and "inline"
embedding of MathML and how this can be specified in particular
instances. In general, a MathML renderer may determine these initial
values in whatever manner is appropriate for the location and context
of the specific instance of MathML it is rendering, or if it has no
way to determine this, based on the way it is most likely to be used;
as a last resort it is suggested that it use the most generic values
displaystyle = ""true"" and
scriptlevel = ""0"".

The MathML layout schemata that typically display some of their
arguments in smaller type or with less vertical spacing, namely the
elements for scripts, fractions, radicals, and tables or matrices,
set displaystyle to "false", and in some cases increase
scriptlevel, for those arguments. The new values are inherited
by all sub-expressions within those arguments, unless they are
overridden.

The specific rules by which each element modifies
displaystyle and/or scriptlevel are given in the
specification for each element that does so; the complete list of
elements that modify either attribute are: the "scripting" elements
msub, msup, msubsup,
munder, mover,
munderover, and mmultiscripts; and the
elements mfrac, mroot, and
mtable.

When mstyle is given a
scriptlevel attribute with no sign, it sets the value of
scriptlevel within its contents to the value given, which
must be a nonnegative integer. When the attribute value consists of a
sign followed by an integer, the value of scriptlevel is
incremented (for '+') or decremented (for '-') by the amount
given. The incremental syntax for this attribute is an exception to
the general rules for setting inherited attributes using
mstyle, and is not allowed by any other attribute
on mstyle.

Whenever the scriptlevel is changed, either
automatically or by being explicitly incremented, decremented, or set,
the current font size is multiplied by the value of
scriptsizemultiplier to the power of the change in
scriptlevel. For example, if scriptlevel is
increased by 2, the font size is multiplied by
scriptsizemultiplier twice in succession; if
scriptlevel is explicitly set to 2 when it had been 3,
the font size is divided by scriptsizemultiplier.
References to fontsize in this section should be
interpreted to mean either the fontsize attribute
or the mathsize attribute.

The default value of scriptsizemultiplier is less than
one (in fact, it is approximately the square root of 1/2), resulting
in a smaller font size with increasing scriptlevel. To
prevent scripts from becoming unreadably small, the font size is never
allowed to go below the value of scriptminsize as a
result of a change to scriptlevel, though it can be set
to a lower value using the fontsize attribute (Section 3.2.2 Mathematics style attributes common to token
elements) on mstyle or on token
elements. If a change to scriptlevel would cause the font
size to become lower than scriptminsize using the above
formula, the font size is instead set equal to
scriptminsize within the sub-expression for which
scriptlevel was changed.

In the syntax for scriptminsize, v-unit represents a unit of vertical length (as
described in Section 2.4.4.2 Attributes with units). The most common unit for specifying font sizes
in typesetting is pt (points).

Explicit changes to the fontsize attribute have no
effect on the value of scriptlevel.

3.3.4.2.2 Further details on scriptlevel for renderers

For MathML renderers that support CSS style sheets, or some other
analogous style sheet mechanism, absolute or relative changes to
fontsize (or other attributes) may occur implicitly on
any element in response to a style sheet. Changes to
fontsize of this kind also have no effect on
scriptlevel. A style sheet-induced change to
fontsize overrides scriptminsize in the same
way as for an explicit change to fontsize in the
element's start tag (discussed above), whether it is specified in the
style sheet as an absolute or a relative change. (However, any
subsequent scriptlevel-induced change to
fontsize will still be affected by it.) As is required
for inherited attributes in CSS, the style sheet-modified
fontsize is inherited by child elements.

If the same element is subject to both a style sheet-induced and an
automatic (scriptlevel-related) change to its own
fontsize, the scriptlevel-related change is
done first - in fact, in the simplest implementation of the
element-specific rules for scriptlevel, this change would
be done by the element's parent as part of producing the rendering
properties it passes to the given element, since it is the parent
element that knows whether scriptlevel should be changed
for each of its child elements.

If the element's own fontsize is changed by a style
sheet and it also changes scriptlevel (and thus
fontsize) for one of its children, the style
sheet-induced change is done first, followed by the change inherited
by that child. If more than one child's scriptlevel is
changed, the change inherited by each child has no effect on the other
children. (As a mnemonic rule that applies to a "parse tree" of
elements and their children, style sheet-induced changes to
fontsize can be associated to nodes of the tree, i.e. to
MathML elements, and scriptlevel-related changes can be
associated to the edges between parent and child elements; then the
order of the associated changes corresponds to the order of nodes and
edges in each path down the tree.) For general information on the
relative order of processing of properties set by style sheets versus by
attributes, see the appropriate subsection of CSS-compatible
attributes in Section 2.4.4.3 CSS-compatible attributes.

If scriptlevel is changed incrementally by an
mstyle element that also sets certain other
attributes, the overall effect of the changes may depend on the order
in which they are processed. In such cases, the attributes in the
following list should be processed in the following order, regardless
of the order in which they occur in the XML-format attribute list of
the mstyle start tag:
scriptsizemultiplier, scriptminsize,
scriptlevel, fontsize.

Note that scriptlevel can, in principle, attain any
integral value by being decremented sufficiently, even though it can
only be explicitly set to nonnegative values. Negative values of
scriptlevel generated in this way are legal and should
work as described, generating font sizes larger than those of the
surrounding expression. Since scriptlevel is initially 0
and never decreases automatically, it will always be nonnegative
unless it is decremented past 0 using mstyle.

Explicit decrements of scriptlevel after the font size
has been limited by scriptminsize as described above
would produce undesirable results. This might occur, for example, in a
representation of a continued fraction, in which the scriptlevel was
decremented for part of the denominator back to its value for the
fraction as a whole, if the continued fraction itself was located in a
place that had a high scriptlevel. To prevent this
problem, MathML renderers should, when decrementing
scriptlevel, use as the initial font size the value the
font size would have had if it had never been limited by
scriptminsize. They should not, however, ignore the
effects of explicit settings of fontsize, even to values
below scriptminsize.

Since MathML renderers may be unable to make use of arbitrary font
sizes with good results, they may wish to modify the mapping from
scriptlevel to fontsize to produce better renderings in their
judgment. In particular, if fontsizes have to be rounded to available
values, or limited to values within a range, the details of how this
is done are up to the renderer. Renderers should, however, ensure that
a series of incremental changes to scriptlevel resulting in its
return to the same value for some sub-expression that it had in a
surrounding expression results in the same fontsize for that
sub-expression as for the surrounding expression.

3.3.4.2.3 Color and background attributes

3.3.4.2.4 Precise background region not specified

The suggested MathML visual rendering rules do not define the
precise extent of the region whose background is affected by using the
background attribute on mstyle,
except that, when mstyle's content does not have
negative dimensions and its drawing region is not overlapped by other
drawing due to surrounding negative spacing, this region should lie
behind all the drawing done to render the content of the
mstyle, but should not lie behind any of the
drawing done to render surrounding expressions. The effect of overlap
of drawing regions caused by negative spacing on the extent of the
region affected by the background attribute is not
defined by these rules.

3.3.4.2.5 Meaning of named mathspaces

The spacing between operators is often one of a small number of
potential values. MathML names these values and allows their values to
be changed. Because the default values for spacing around operators
that are given in the operator dictionary Appendix F Operator Dictionary
are defined using these named spaces, changing their values will produce
tighter or looser spacing. These values can be used anywhere a h-unit or v-unit unit is
allowed. See Section 2.4.4.2 Attributes with units.

3.3.5 Error Message (merror)

3.3.5.1 Description

The merror element displays its contents as an
"error message". This might be done, for example, by displaying the
contents in red, flashing the contents, or changing the background
color. The contents can be any expression or expression sequence.

merror accepts any number of arguments; if
this number is not 1, its contents are treated as a single "inferred
mrow" as described in Section 3.1.3 Required Arguments.

The intent of this element is to provide a standard way for
programs that generate MathML from other input to report
syntax errors in their input. Since it is anticipated that
preprocessors that parse input syntaxes designed for easy hand entry
will be developed to generate MathML, it is important that they have
the ability to indicate that a syntax error occurred at a certain
point. See Section 7.2.2 Handling of Errors.

The suggested use of merror for reporting
syntax errors is for a preprocessor to replace the erroneous part of
its input with an merror element containing a
description of the error, while processing the surrounding expressions
normally as far as possible. By this means, the error message will be
rendered where the erroneous input would have appeared, had it been
correct; this makes it easier for an author to determine from the
rendered output what portion of the input was in error.

No specific error message format is suggested here, but as with
error messages from any program, the format should be designed to make
as clear as possible (to a human viewer of the rendered error message)
what was wrong with the input and how it can be fixed. If the
erroneous input contains correctly formatted subsections, it may be
useful for these to be preprocessed normally and included in the error
message (within the contents of the merror
element), taking advantage of the ability of
merror to contain arbitrary MathML expressions
rather than only text.

Note that the preprocessor's input is not, in this case, valid MathML,
but the error message it outputs is valid MathML.

3.3.6 Adjust Space Around Content
(mpadded)

3.3.6.1 Description

An mpadded element renders the same as its
content, but with its overall size and other dimensions (such as
baseline position) modified according to its attributes. The
mpadded element does not rescale (stretch or
shrink) its content; its only effect is to modify the apparent size
and position of the "bounding box" around its content, so as to
affect the relative position of the content with respect to the
surrounding elements. The name of the element reflects the use of
mpadded to effectively add "padding", or extra
space, around its content. If the "padding" is negative, it is
possible for the content of mpadded to be
rendered outside the mpadded element's bounding
box; see below for warnings about several potential pitfalls of this
effect.

The mpadded element accepts any number of
arguments; if this number is not 1, its contents are treated as a single
"inferred mrow" as described in
Section 3.1.3 Required Arguments.

It is suggested that audio renderers add (or shorten) time delays
based on the attributes representing horizontal space
(width and lspace).

These attributes modify the dimensions of the "bounding
box" of the mpadded element. The dimensions
(which have the same names as the attributes) are defined in the next
subsection. Depending on the format of the attribute value, a dimension
may be set to a new value, or to an incremented or decremented version of
the content's corresponding dimension. Values may be specified as multiples
or percentages of any of the dimensions of the normal rendering of the
element's content (using so-called "pseudo-units"), or
they can be set directly using standard units Section 2.4.4.2 Attributes with units.

If an attribute value begins with a + or
- sign, it specifies an increment or decrement of the
corresponding dimension by the following length value (interpreted as
explained below). Otherwise, the corresponding dimension is set
directly to the following length value. Note that the +
and - do not mean that the following value is positive or
negative, even when an explicit length unit (h-unit or
v-unit) is given. In particular, these attributes cannot
directly set a dimension to a negative value.

Length values (after the optional sign, which is not part of the
length value) can be specified in several formats. Each format begins
with an unsigned-number, which may be followed by a
% sign and an optional "pseudo-unit" (denoted by
pseudo-unit in the attribute syntaxes above), by a
pseudo-unit alone, or by one of the length units
(denoted by h-unit or v-unit) specified in
Section 2.4.4.2 Attributes with units, not including %. The possible
pseudo-units are the keywords width, lspace,
height, and depth; they each represent the
length of the same-named dimension of the mpadded
element's content (not of the mpadded element
itself). The lengths represented by h-unit or
v-unit are described in Section 2.4.4.2 Attributes with units.

In any of these formats, the length value specified is the product
of the specified number and the length represented by the unit or
pseudo-unit. The result is multiplied by 0.01 if % is given. If no
pseudo-unit is given after %, the one with the same name
as the attribute being specified is assumed.

Some examples of attribute formats using pseudo-units (explicit or
default) are as follows: depth="100% height" and
depth="1.0 height" both set the depth of the
mpadded element to the height of its content.
depth="105%" sets the depth to 1.05 times the content's
depth, and either depth="+100%" or
depth="200%" sets the depth to twice the content's
depth.

Dimensions that would be positive if the content was rendered
normally cannot be made negative using mpadded; a
positive dimension is set to 0 if it would otherwise become negative.
Dimensions that are initially 0 can be made negative, but this
should generally be avoided. See the warnings below on the use of
negative spacing for "tweaking" or conveying meaning.

The rules given above imply that all of the following attribute
settings have the same effect, which is to leave the content's
dimensions unchanged:

3.3.6.3 Meanings of dimension attributes

The width attribute refers to the overall horizontal
width of a bounding box. By default (i.e. when lspace is
not modified), the bounding box of the content of an
mpadded element should be rendered flush with the
left edge of the mpadded element's bounding
box. Thus, increasing width alone effectively adds space
on the right edge of the box.

The lspace attribute refers to the amount of space
between the left edge of a bounding box and the start of the rendering of its
contents' bounding box. Unlike the other dimensions,
lspace does not correspond to a real property of a
bounding box, but exists only transiently during the computations done
by each instance of mpadded. It is provided so
that there is a way to add space on the left edge of a bounding
box.

The rationale behind using width and
lspace to control horizontal padding instead of more
symmetric attributes, such as a hypothetical rspace and
lspace, is that it is desirable to have a "width" pseudo
unit, in part because "width" is an actual property of a bounding
box.

The height attribute refers to the amount of vertical
space between the baseline (the line along the bottom of most letter
glyphs in normal text rendering) and the top of the bounding box.

The depth attribute refers to the amount of vertical
space between the bottom of the bounding box and the baseline.

MathML renderers should ensure that, except for the effects of the
attributes, relative spacing between the contents of
mpadded and surrounding MathML elements is not
modified by replacing an mpadded element with an
mrow element with the same content. This holds
even if linebreaking occurs within the mpadded
element. However, if an mpadded element with
non-default attribute values is subjected to linebreaking, MathML does
not define how its attributes or rendering interact with the
linebreaking algorithm.

3.3.6.4 Warning: nonportability of "tweaking"

A likely temptation for the use of the mpadded
and mspace elements (and perhaps also mphantom and mtext) will be
for an author to improve the spacing generated by a specific renderer by
slightly modifying it in specific expressions, i.e. to
"tweak" the rendering.

Authors are strongly warned that different MathML renderers
may use different spacing rules for computing the relative
positions of rendered symbols in expressions that have no explicit
modifications to their spacing; if renderer B improves upon renderer
A's spacing rules, explicit spacing added to improve the output
quality of renderer A may produce very poor results in renderer B,
very likely worse than without any "tweaking" at all.

Even when a specific choice of renderer can be assumed, its spacing
rules may be improved in successive versions, so that the effect of
tweaking in a given MathML document may grow worse with time. Also,
when style sheet mechanisms are extended to MathML, even one version
of a renderer may use different spacing rules for users with different
style sheets.

Therefore, it is suggested that MathML markup never use
mpadded or mspace elements
to tweak the rendering of specific expressions, unless the MathML is
generated solely to be viewed using one specific version of one MathML
renderer, using one specific style sheet (if style sheets are
available in that renderer).

In cases where the temptation to improve spacing proves too strong,
careful use of mpadded,
mphantom, or the alignment elements (Section 3.5.5 Alignment Markers) may give more portable results than the
direct insertion of extra space using mspace or
mtext. Advice given to the implementors of MathML
renderers might be still more productive, in the long run.

3.3.6.5 Warning: spacing should not be used to convey meaning

MathML elements that permit "negative spacing", namely
mspace, mpadded, and
mtext, could in theory be used to simulate new
notations or "overstruck" characters by the visual overlap of the
renderings of more than one MathML sub-expression.

This practice is strongly discouraged in all situations,
for the following reasons:

it will give different results in different MathML renderers
(so the warning about "tweaking" applies), especially
if attempts are made to render glyphs outside the bounding box of
the MathML expression;

it is likely to appear much worse than a more standard construct
supported by good renderers;

such expressions are almost certain to be uninterpretable
by audio renderers, computer algebra systems,
text searches for standard symbols,
or other processors of MathML input.

More generally, any construct that uses spacing to convey
mathematical meaning, rather than simply as an aid to viewing
expression structure, is discouraged. That is, the constructs that
are discouraged are those that would be interpreted differently by a
human viewer of rendered MathML if all explicit spacing was
removed.

If such constructs are used in spite of this warning, they should
be enclosed in a semantics element that also
provides an additional MathML expression that can be interpreted in a
standard way.

forms an overstruck symbol in violation of the policy stated above;
it might be intended to represent the set of complex numbers for a
MathML renderer that lacks support for the standard symbol used for
this purpose. This kind of construct should always be avoided in
MathML, for the reasons stated above; indeed, it should never be
necessary for standard symbols, since a MathML renderer with no better
method of rendering them is free to use overstriking internally, so
that it can still support general MathML input.

However, if for whatever reason such a construct is used in MathML,
it should always be enclosed in a semantics element such as

which provides an alternative, standard encoding for the desired
symbol, which is much more easily interpreted than the construct using
negative spacing. The alternative encoding in this example uses
MathML presentation elements; the content elements described in
Chapter 4 Content Markup should also be considered.

The above warning also applies to most uses of rendering
attributes to alter the meaning conveyed by an expression, with the
exception of attributes on mi (such as
fontweight) used to distinguish one variable from
another.

3.3.7 Making Sub-Expressions Invisible (mphantom)

3.3.7.1 Description

The mphantom element renders invisibly, but
with the same size and other dimensions, including baseline position,
that its contents would have if they were rendered
normally. mphantom can be used to align parts of
an expression by invisibly duplicating sub-expressions.

The mphantom element accepts any number of
arguments; if this number is not 1, its contents are treated as a single
"inferred mrow" formed from all its
arguments, as described in Section 3.1.3 Required Arguments.

3.3.7.2 Attributes

Note that it is possible to wrap both an
mphantom and an mpadded
element around one MathML expression, as in
<mphantom><mpadded attribute-settings>
... </mpadded></mphantom>, to change its size and make it
invisible at the same time.

MathML renderers should ensure that the relative spacing between
the contents of an mphantom element and the
surrounding MathML elements is the same as it would be if the
mphantom element were replaced by an
mrow element with the same content. This holds
even if linebreaking occurs within the mphantom
element.

For the above reason, mphantom is
not considered space-like (Section 3.2.7 Space (mspace)) unless its
content is space-like, since the suggested rendering rules for
operators are affected by whether nearby elements are space-like. Even
so, the warning about the legal grouping of space-like elements may
apply to uses of mphantom.

There is one situation where the preceding rule for rendering an
mphantom may not give the desired effect. When an
mphantom is wrapped around a subsequence of the
arguments of an mrow, the default determination
of the form attribute for an mo
element within the subsequence can change. (See the default value of
the form attribute described in Section 3.2.5 Operator, Fence, Separator or Accent
(mo).) It may be
necessary to add an explicit form attribute to such an
mo in these cases. This is illustrated in the
following example.

3.3.7.3 Examples

In this example, mphantom is used to ensure
alignment of corresponding parts of the numerator and denominator of a
fraction:

The explicit attribute setting form="infix" on the
mo element inside the mphantom sets the
form attribute to what it would have been in the absence of the
surrounding mphantom. This is necessary since
otherwise, the + sign would be interpreted as a prefix
operator, which might have slightly different spacing.

Alternatively, this problem could be avoided without any explicit
attribute settings, by wrapping each of the arguments
<mo>+</mo> and <mi>y</mi> in its
own mphantom element, i.e.

3.3.8 Expression Inside Pair of Fences
(mfenced)

3.3.8.1 Description

The mfenced element provides a convenient form
in which to express common constructs involving fences (i.e. braces,
brackets, and parentheses), possibly including separators (such as
comma) between the arguments.

For example, <mfenced> <mi>x</mi> </mfenced>
renders as "(x)" and is equivalent to

<mrow> <mo> ( </mo> <mi>x</mi> <mo> ) </mo> </mrow>

and
<mfenced> <mi>x</mi> <mi>y</mi> </mfenced>
renders as "(x, y)"
and is equivalent to

Individual fences or separators are represented using
mo elements, as described in Section 3.2.5 Operator, Fence, Separator or Accent
(mo). Thus, any mfenced
element is completely equivalent to an expanded form described below;
either form can be used in MathML, at the convenience of an author or
of a MathML-generating program. A MathML renderer is required to
render either of these forms in exactly the same way.

In general, an mfenced element can contain
zero or more arguments, and will enclose them between fences in an
mrow; if there is more than one argument, it will
insert separators between adjacent arguments, using an additional
nested mrow around the arguments and separators
for proper grouping (Section 3.3.1 Horizontally Group Sub-Expressions
(mrow)). The general expanded form is
shown below. The fences and separators will be parentheses and comma
by default, but can be changed using attributes, as shown in the
following table.

The "opening-fence" and "closing-fence" are
arbitrary strings. (Since they are used as the content of
mo elements, any whitespace they contain will be
trimmed and collapsed as described in Section 2.4.6 Collapsing Whitespace in Input.)

The value of separators is a sequence of zero or more
separator characters (or entity references), optionally separated by
whitespace. Each sep#i consists of exactly
one character or entity reference. Thus, separators=",;"
is equivalent to separators=" , ; ".

The general mfenced element shown above is
equivalent to the following expanded form:

If there are too many separator characters, the extra ones are
ignored. If separator characters are given, but there are too few, the
last one is repeated as necessary. Thus, the default value of
separators="," is equivalent to
separators=",,", separators=",,,", etc. If
there are no separator characters provided but some are needed, for
example if separators=" " or "" and there is more than
one argument, then no separator elements are inserted at all - that
is, the elements <mo separator="true"> sep#i
</mo> are left out entirely. Note that this is different
from inserting separators consisting of mo
elements with empty content.

Note that not all "fenced expressions" can be encoded by an
mfenced element. Such exceptional expressions
include those with an "embellished" separator or fence or one
enclosed in an mstyle element, a missing or extra
separator or fence, or a separator with multiple content
characters. In these cases, it is necessary to encode the expression
using an appropriately modified version of an expanded form. As
discussed above, it is always permissible to use the expanded form
directly, even when it is not necessary. In particular, authors cannot
be guaranteed that MathML preprocessors won't replace occurrences of
mfenced with equivalent expanded forms.

Note that the equivalent expanded forms shown above include
attributes on the mo elements that identify them as fences or
separators. Since the most common choices of fences and separators
already occur in the operator dictionary with those attributes,
authors would not normally need to specify those attributes explicitly
when using the expanded form directly. Also, the rules for the default
form attribute (Section 3.2.5 Operator, Fence, Separator or Accent
(mo)) cause the
opening and closing fences to be effectively given the values
form="prefix" and
form="postfix" respectively, and the
separators to be given the value
form="infix".

Note that it would be incorrect to use mfenced
with a separator of, for instance, "+", as an abbreviation for an
expression using "+" as an ordinary operator, e.g.

<mrow>
<mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi>
</mrow>

This is because the + signs would be treated as separators,
not infix operators. That is, it would render as if they were marked up as
<mo separator="true">+</mo>, which might therefore
render inappropriately.

3.3.9 Enclose Expression Inside Notation
(menclose)

3.3.9.1 Description

The menclose element renders its content
inside the enclosing notation specified by its notation attribute.
menclose accepts any number of arguments; if
this number is not 1, its contents are treated as a single "inferred
mrow" containing its arguments,
as described in Section 3.1.3 Required Arguments.

When notation has the value "longdiv", the contents are drawn enclosed by a long
division symbol. A complete example of long division is accomplished
by also using mtable and malign. When notation is
specified as "actuarial", the contents are drawn
enclosed by an actuarial symbol. The case of notation="radical" is
equivalent to the msqrt schema.

3.3.9.3 Examples

The following markup might be used to encode an elementary US-style
long division problem.

3.4 Script and Limit Schemata

The elements described in this section position one or more scripts
around a base. Attaching various kinds of scripts and embellishments to
symbols is a very common notational device in mathematics. For purely
visual layout, a single general-purpose element could suffice for
positioning scripts and embellishments in any of the traditional script
locations around a given base. However, in order to capture the abstract
structure of common notation better, MathML provides several more
specialized scripting elements.

In addition to sub/superscript elements, MathML has overscript
and underscript elements that place scripts above and below the base. These
elements can be used to place limits on large operators, or for placing
accents and lines above or below the base. The rules for rendering accents
differ from those for overscripts and underscripts, and this difference can
be controlled with the accent and accentunder attributes, as described in the appropriate
sections below.

Rendering of scripts is affected by the scriptlevel and displaystyle
attributes, which are part of the environment inherited by the rendering
process of every MathML expression, and are described under mstyle (Section 3.3.4 Style Change (mstyle)). These
attributes cannot be given explicitly on a scripting element, but can be
specified on the start tag of a surrounding mstyle
element if desired.

MathML also provides an element for attachment of tensor indices.
Tensor indices are distinct from ordinary subscripts and superscripts in
that they must align in vertical columns. Tensor indices can also occur in
prescript positions.

Because presentation elements should be used to describe the abstract
notational structure of expressions, it is important that the base
expression in all "scripting" elements (i.e. the first
argument expression) should be the entire expression that is being
scripted, not just the rightmost character. For example,
(x+y)2 should be written as:

The msub element increments
scriptlevel by 1, and sets displaystyle to
"false", within subscript, but leaves both attributes
unchanged within base. (These attributes are inherited by
every element through its rendering environment, but can be set
explicitly only on mstyle; see Section 3.3.4 Style Change (mstyle).)

The msup element increments scriptlevel by 1, and sets displaystyle to "false", within
superscript, but leaves both attributes unchanged within
base. (These attributes are inherited by every element through
its rendering environment, but can be set explicitly only on mstyle; see Section 3.3.4 Style Change (mstyle).)

3.4.3 Subscript-superscript Pair (msubsup)

3.4.3.1 Description

The msubsup element is used to attach both a subscript and
superscript to a base expression. Note that both scripts are
positioned tight against the base as shown here
versus the staggered positioning of nested scripts as shown here
.

The msubsup element increments
scriptlevel by 1, and sets displaystyle to
"false", within subscript and superscript,
but leaves both attributes unchanged within base. (These
attributes are inherited by every element through its rendering
environment, but can be set explicitly only on
mstyle; see Section 3.3.4 Style Change (mstyle).)

3.4.3.3 Examples

The msubsup is most commonly used for adding
sub/superscript pairs to identifiers as illustrated above. However,
another important use is placing limits on certain large operators
whose limits are traditionally displayed in the script positions even
when rendered in display style. The most common of these is the
integral. For example,

The accentunder attribute controls whether
underscript is drawn as an "accent" or as a limit. The
main difference between an accent and a limit is that the limit is
reduced in size whereas an accent is the same size as the base. A
second difference is that the accent is drawn closer to the base.

The default value of accentunder is false, unless
underscript is an mo element or an
embellished operator (see Section 3.2.5 Operator, Fence, Separator or Accent
(mo)). If
underscript is an mo element, the
value of its accent attribute is used as the default
value of accentunder. If underscript is an
embellished operator, the accent attribute of the
mo element at its core is used as the default
value. As with all attributes, an explicitly given value overrides
the default.

Here is an example (accent versus underscript):
versus
.
The MathML representation for this example is shown below.

If the base is an operator with movablelimits="true" (or an embellished operator whose mo element core has movablelimits="true"), and displaystyle="false", then
underscript is drawn in a subscript position. In this case,
the accentunder attribute is ignored. This is often
used for limits on symbols such as &sum;.

Within underscript, munder always
sets displaystyle to "false", but increments
scriptlevel by 1 only when accentunder is
"false". Within base, it always leaves both attributes
unchanged. (These attributes are inherited by every element through
its rendering environment, but can be set explicitly only on
mstyle; see Section 3.3.4 Style Change (mstyle).)

The accent attribute controls whether
overscript is drawn as an "accent" (diacritical mark) or
as a limit. The main difference between an accent and a limit is that
the limit is reduced in size whereas an accent is the same size as the
base. A second difference is that the accent is drawn closer to the
base. This is shown below (accent versus limit):
versus
.

These differences also apply to "mathematical accents" such as
bars or braces over expressions:
versus
.
The MathML representation for each of these examples is shown below.

The default value of accent is false, unless
overscript is an mo element or an
embellished operator (see Section 3.2.5 Operator, Fence, Separator or Accent
(mo)). If
overscript is an mo element, the value
of its accent attribute is used as the default value
of accent for mover. If
overscript is an embellished operator, the accent attribute of the mo
element at its core is used as the default value.

If the base is an operator with movablelimits="true" (or an embellished operator whose mo element core has movablelimits="true"), and displaystyle="false", then
overscript is drawn in a superscript position. In this case,
the accent attribute is ignored. This is often used
for limits on symbols such as &sum;.

Within overscript, mover always
sets displaystyle to "false",
but increments scriptlevel by 1 only when accent is "false". Within
base, it always leaves both attributes unchanged. (These
attributes are inherited by every element through its rendering
environment, but can be set explicitly only on mstyle; see Section 3.3.4 Style Change (mstyle).)

The munderover element is used so that the
underscript and overscript are vertically spaced equally in relation
to the base and so that they follow the slant of the base as in the
second expression shown below:

versus

The MathML representation for this example is shown below.

The difference in the vertical spacing is too small to be noticed on a
low resolution display at a normal font size, but is noticeable on a higher
resolution device such as a printer and when using large font sizes. In
addition to the visual differences, attaching both the underscript and
overscript to the same base more accurately reflects the semantics of the
expression.

The accent and accentunder
attributes have the same effect as the attributes with the same names on
mover (Section 3.4.5 Overscript (mover)) and munder (Section 3.4.4 Underscript (munder)),
respectively. Their default values are also computed in the same manner as
described for those elements, with the default value of accent depending on overscript and the
default value of accentunder depending on
underscript.

If the base is an operator with movablelimits="true" (or an embellished operator whose mo element core has movablelimits="true"), and displaystyle="false", then
underscript and overscript are drawn in a
subscript and superscript position, respectively. In this case, the accent and accentunder attributes
are ignored. This is often used for limits on symbols such as &sum;.

Within underscript, munderover
always sets displaystyle to "false", but increments scriptlevel
by 1 only when accentunder is "false". Within overscript, munderover always sets displaystyle to "false", but
increments scriptlevel by 1 only when accent is "false". Within
base, it always leaves both attributes unchanged. (These
attributes are inherited by every element through its rendering
environment, but can be set explicitly only on mstyle; see Section 3.3.4 Style Change (mstyle)).

3.4.6.3 Examples

The MathML representation for the example shown above with the first
expression made using separate munder and
mover elements, and the second one using an
munderover element, is:

3.4.7 Prescripts and Tensor Indices
(mmultiscripts)

3.4.7.1 Description

Presubscripts and tensor notations are represented by a single
element, mmultiscripts. This element allows the
representation of any number of vertically-aligned pairs of subscripts
and superscripts, attached to one base expression. It supports both
postscripts (to the right of the base in visual notation) and
prescripts (to the left of the base in visual notation). Missing
scripts can be represented by the empty element
none.

The prescripts are optional, and when present are given
after the postscripts, because prescripts are relatively
rare compared to tensor notation.

The argument sequence consists of the base followed by zero or more
pairs of vertically-aligned subscripts and superscripts (in that
order) that represent all of the postscripts. This list is optionally
followed by an empty element mprescripts and a
list of zero or more pairs of vertically-aligned presubscripts and
presuperscripts that represent all of the prescripts. The pair lists
for postscripts and prescripts are given in a left-to-right order. If
no subscript or superscript should be rendered in a given position,
then the empty element none should be used in
that position.

The base, subscripts, superscripts, the optional separator element
mprescripts, the presubscripts, and the
presuperscripts, are all direct sub-expressions of the
mmultiscripts element, i.e. they are all at the
same level of the expression tree. Whether a script argument is a
subscript or a superscript, or whether it is a presubscript or a
presuperscript is determined by whether it occurs in an even-numbered
or odd-numbered argument position, respectively, ignoring the empty
element mprescripts itself when determining the
position. The first argument, the base, is considered to be in
position 1. The total number of arguments must be odd, if
mprescripts is not given, or even, if it is.

The empty elements mprescripts and
none are only allowed as direct sub-expressions
of mmultiscripts.

3.4.7.2 Attributes

The mmultiscripts element increments scriptlevel by 1, and sets displaystyle to "false", within
each of its arguments except base, but leaves both attributes
unchanged within base. (These attributes are inherited by
every element through its rendering environment, but can be set explicitly
only on mstyle; see Section 3.3.4 Style Change (mstyle).)

3.5 Tables and Matrices

Matrices, arrays and other table-like mathematical notation are marked
up using mtable,
mtr, mlabeledtr and
mtd elements. These elements are similar to the
table, tr and td elements of HTML, except that they provide
specialized attributes for the fine layout control
necessary for commutative diagrams, block matrices and so on.

The mlabeledtr element represents a labeled
row of a table and can be used for numbered equations.
The first child of mlabeledtr is the label.
A label is somewhat special in that it is not considered an expression
in the matrix and is not counted when determining the number of columns
in that row.

3.5.1 Table or Matrix
(mtable)

3.5.1.1 Description

A matrix or table is specified using the mtable element. Inside of the mtable element, only mtr
or mlabeledtr elements may appear.

In MathML 1.x, the mtable element could
infer mtr elements around its arguments, and
the mtr element could infer mtd elements. In other words, if some argument to
an mtable was not an mtr element, a MathML application was to assume a
row with a single column (i.e. the argument was effectively wrapped
with an inferred mtr). Similarly, if some
argument to a (possibly inferred) mtr element
was not an mtd element, that argument was to
be treated as a table entry by wrapping it with an inferred mtd element.
MathML 2.0 deprecates the inference
of mtr and mtd elements;
mtr and mtd elements
must be used inside of mtable and
mtr respectively.

Table rows that have fewer columns than other rows of the same
table (whether the other rows precede or follow them) are effectively
padded on the right with empty mtd elements so
that the number of columns in each row equals the maximum number of
columns in any row of the table. Note that the use of
mtd elements with non-default values of the
rowspan or columnspan
attributes may affect
the number of mtd elements that should be given
in subsequent mtr elements to cover a given
number of columns.
Note also that the label in an mlabeledtr element
is not considered a column in the table.

Note that the default value for each of rowlines, columnlines and
frame is the literal string
"none", meaning that the default is to render no lines,
rather than that there is no default.

As described in Section 2.4.4 MathML Attribute Values, the notation (x
| y)+ means one or more occurrences of either x or
y, separated by whitespace. For example, possible values
for columnalign are "left", "left left", and "left right center center". If there are more
entries than are necessary (e.g. more entries than columns for columnalign), then only the first entries will be
used. If there are fewer entries, then the last entry is repeated as
often as necessary. For example, if columnalign="right center" and the table has three
columns, the first column will be right aligned and the second and
third columns will be centered. The label in a mlabeledtr is not considered as a column in the
table and the attribute values that apply to columns do not apply to
labels.

The align attribute specifies where to align the
table with respect to its environment. "axis" means to align
the center of the table on the environment's axis. (The axis of an equation
is an alignment line used by typesetters. It is the line on which a minus
sign typically lies. The center of the table is the midpoint of the
table's vertical extent.) "center" and "baseline"
both mean to align the center of the table on the environment's
baseline. "top" or "bottom" aligns the top or
bottom of the table on the environment's baseline.

If the align attribute value ends with a
"rownumber" between 1 and n (for a table with
n rows), the specified row is aligned in the way described above,
rather than the table as a whole; the top (first) row is numbered 1, and
the bottom (last) row is numbered n. The same is true if the
row number is negative, between -1 and -n,
except that the bottom row is referred to as -1 and the top row as
-n. Other values of "rownumber" are
illegal.

The rowalign attribute specifies how the entries in
each row should be aligned. For example, "top" means that the tops of
each entry in each row should be aligned with the tops of the other
entries in that row. The columnalign attribute specifies
how the entries in each column should be aligned.

The columnwidth attribute specifies how wide
a column should be. The "auto" value means that
the column should be as wide as needed, which is the default. If an
explicit value is given, then the column is exactly that wide and the
contents of that column are made to fit in that width. The contents
are linewrapped or clipped at the discretion of the renderer. If "fit" is given as a value, the remaining page width
after subtracting the widths for columns specified as "auto" and/or specific widths is divided equally
among the "fit" columns and this value is used
for the column width. If insufficient room remains to hold the
contents of the "fit" columns, renderers may
linewrap or clip the contents of the "fit"
columns. When the columnwidth is specified as
a percentage, the value is relative to the width of the table. That
is, a renderer should try to adjust the width of the column so that it
covers the specified percentage of the entire table width.

The width attribute specifies the desired
width of the entire table and is intended for visual user agents. When
the value is a percentage value, the value is relative to the
horizontal space a MathML renderer has available for the math
element. When the value is "auto", the MathML
renderer should calculate the table width from its contents using
whatever layout algorithm it chooses.

MathML 2.0 does not specify a table layout algorithm. In
particular, it is the responsibility of a MathML renderer to resolve
conflicts between the width attribute and other
constraints on the width of a table, such as explicit values for columnwidth attributes, and minimum sizes for table
cell contents. For a discussion of table layout algorithms, see
Cascading
Style Sheets, level 2.

The rowspacing and columnspacing
attributes specify how much space should be added between each row and
column. However, spacing before the first row and after the last row
(i.e. at the top and bottom of the table) is given by the second
number in the value of the framespacing attribute, and
spacing before the first column and after the last column (i.e. on the
left and on the right of the table) is given by the first number in
the value of the framespacing attribute.

In those attributes' syntaxes, h-unit or
v-unit represents a unit of horizontal or vertical
length, respectively (see Section 2.4.4.2 Attributes with units). The units shown in the
attributes' default values (em or ex) are
typically used.

The rowlines and columnlines attributes
specify whether and what kind of lines should be added between each
row and column. Lines before the first row or column and after the
last row or column are given using the frame
attribute.

If a frame is desired around the table, the frame
attribute is used. If the attribute value is not "none", then
framespacing is used to add spacing between the lines of
the frame and the first and last rows and columns of the table. If
frame="none", then the framespacing
attribute is ignored. The frame and
framespacing attributes are not part of the
rowlines/columnlines,
rowspacing/columnspacing options because
having them be so would often require that rowlines and
columnlines would need to be fully specified instead of
just giving a single value.
For example, if a table had five columns and it was desired to have
no frame around the table but to have lines between the columns, then
columnlines="none solid solid solid solid none"
would be necessary. If the frame is separated from the internal
lines, only columnlines="solid" is needed.

The equalrows attribute forces the rows all to be
the same total height when set to "true". The equalcolumns attribute forces the columns all to be the
same width when set to "true".

The displaystyle attribute specifies the
value of displaystyle (described under mstyle in Section 3.3.4 Style Change (mstyle)) within
each cell (mtd element) of the table. Setting
displaystyle="true" can be
useful for tables whose elements are whole mathematical expressions;
the default value of "false" is appropriate when
the table is part of an expression, for example, when it represents a
matrix. In either case, scriptlevel (Section 3.3.4 Style Change (mstyle)) is not changed for the table cells.

The side attribute
specifies what side of a table a label for a table row should should be
placed. This attribute is intended to be used for labeled expressions.
If "left" or "right" is specified,
the label is placed on the left or right side of the table row respectively.
The other two attribute values are variations on
"left" and "right":
if the labeled row fits within the width allowed for the table without
the label,
but does not fit within the width if the label is included, then the
label overlaps the row and is displayed above the row if
rowalign for that row is "top";
otherwise the label is displayed below the row.

If there are multiple labels in a table, the alignment of the labels within
the virtual column that they form is left-aligned for labels on the left
side of the table, and right-aligned for labels on the right side of the
table. The alignment can be overridden by specifying
columnalignment for a mlabeledtr
element.

The minlabelspacing attribute
specifies the minimum space allowed between a label and the adjacent
entry in the row.

Note that the parentheses must be represented explicitly; they are not
part of the mtable element's rendering. This allows
use of other surrounding fences, such as brackets, or none at all.

3.5.2 Row in Table or Matrix (mtr)

3.5.2.1 Description

An mtr element represents one row in a table
or matrix. An mtr element is only allowed as a
direct sub-expression of an mtable element, and
specifies that its contents should form one row of the table. Each
argument of mtr is placed in a different column
of the table, starting at the leftmost column.

The rowalign and columnalign attributes allow a specific row to
override the alignment specified by the same attributes in the
surrounding mtable element.

As with mtable, if there are more entries than
necessary in the value of columnalign (i.e. more entries
than columns in the row), then the extra entries will be ignored. If
there are fewer entries than columns, then the last entry will be
repeated as many times as needed.

3.5.3 Labeled Row in Table or Matrix
(mlabeledtr)

3.5.3.1 Description

An mlabeledtr element represents one row in
a table that has a label on either the left or right side, as
determined by the side attribute. The label is
the first child of mlabeledtr. The rest of
the children represent the contents of the row and are identical to
those used for mtr; all of the children except
the first must be mtd elements.

An mlabeledtr element is only allowed as a
direct sub-expression of an mtable element.
Each argument of mlabeledtr except for the first
argument (the label) is placed in a different column
of the table, starting at the leftmost column.

Note that the label element is not considered to be a cell in the
table row. In particular, the label element is not taken into
consideration in the table layout for purposes of width and alignment
calculations. For example, in the case of an mlabeledtr with a label and a single centered mtd child, the child is first centered in the
enclosing mtable, and then the label is
placed. Specifically, the child is not centered in the
space that remains in the table after placing the label.

While MathML 2.0 does not specify an algorithm for placing labels,
implementors of visual renderers may find the following formatting
model useful. To place a label, an implementor might think in terms
of creating a larger table, with an extra column on both ends. The
columnwidth attributes of both these border
columns would be set to "fit" so that they expand
to fill whatever space remains after the inner columns have been laid
out. Finally, depending on the values of side
and minlabelspacing, the label is placed
in whatever border column is appropriate, possibly shifted down if
necessary.

3.5.3.2 Attributes

The attributes for mlabeledtr are the same
as for mtr. Unlike the attributes for the
mtable element, attributes of
mlabeledtr that apply to column elements
also apply to the label. For example, in a one column table,

<mlabeledtr rowalign='top'>

means that the label and other entries in the row are vertically aligned
along their top. To force a particular alignment on the label,
the appropriate attribute would normally be set on the
mtd start tag that surrounds the label content.

3.5.3.3 Equation Numbering

One of the important uses of mlabeledtr is
for numbered equations. In a mlabeledtr, the
label represents the equation number and the elements in the row are
the equation being numbered. The side and minlabelspacing attributes of mtable determine the placement of the equation
number.

In larger documents with many numbered equations, automatic
numbering becomes important. While automatic equation numbering and
automatically resolving references to equation numbers is outside the
scope of MathML, these problems can be addressed by the use of style
sheets or other means. The mlabeledtr construction provides support
for both of these functions in a way that is intended to facilitate
XSLT processing. The mlabeledtr element can be
used to indicate the presence of a numbered equation, and the first
child can be changed to the current equation number, along with
incrementing the global equation number. For cross references, an
id on either the mlabeledtr element or on the first element
itself could be used as a target of any link.

3.5.4 Entry in Table or Matrix (mtd)

3.5.4.1 Description

An mtd element represents one entry, or cell, in a
table or matrix. An mtd element is only
allowed as a direct sub-expression of an mtr
or an mlabeledtr element.

The mtd element accepts any number of
arguments; if this number is not 1, its contents are treated as a single
"inferred mrow" formed from all its
arguments, as described in Section 3.1.3 Required Arguments.

3.5.4.2 Attributes

Name

values

default

rowspan

number

1

columnspan

number

1

rowalign

top | bottom | center | baseline | axis

inherited

columnalign

left | center | right

inherited

groupalign

group-alignment-list

inherited

The rowspan and columnspan attributes
allow a specific matrix element to be treated as if it occupied the
number of rows or columns specified. The interpretation of how this
larger element affects specifying subsequent rows and columns is meant
to correspond with the similar attributes for HTML 4.01 tables.

The rowspan and columnspan attributes
can be used around an mtd element that represents
the label in a mlabeledtr element.
Also, the label of a mlabeledtr element is not
considered to be part of a previous rowspan and
columnspan.

The rowalign and columnalign attributes
allow a specific matrix element to override the alignment specified by
a surrounding mtable or mtr
element.

3.5.5 Alignment Markers

3.5.5.1 Description

Alignment markers are space-like elements (see Section 3.2.7 Space (mspace)) that can be used
to vertically align specified points within a column of MathML
expressions by the automatic insertion of the necessary amount of
horizontal space between specified sub-expressions.

The discussion that follows will use the example of a set of
simultaneous equations that should be rendered with vertical
alignment of the coefficients and variables of each term, by
inserting spacing somewhat like that shown here:

8.44x + 55 y = 0
3.1 x - 0.7y = -1.1

If the example expressions shown above were arranged in a column
but not aligned, they would appear as:

8.44x + 55y = 0
3.1x - 0.7y = -1.1

For audio renderers, it is suggested that the alignment elements
produce the analogous behavior of altering the rhythm of pronunciation
so that it is the same for several sub-expressions in a column, by the
insertion of the appropriate time delays in place of the extra
horizontal spacing described here.

The expressions whose parts are to be aligned (each equation, in the
example above) must be given as the table elements (i.e. as the mtd elements) of one column of an
mtable. To avoid confusion, the term "table
cell" rather than "table element" will be used in the
remainder of this section.

All interactions between alignment elements are limited to the
mtable column they arise in. That is, every
column of a table specified by an mtable element
acts as an "alignment scope" that contains within it all alignment
effects arising from its contents. It also excludes any interaction
between its own alignment elements and the alignment elements inside
any nested alignment scopes it might contain.

The reason mtable columns are used as
alignment scopes is that they are the only general way in MathML to
arrange expressions into vertical columns. Future versions of MathML
may provide an malignscope element that allows
an alignment scope to be created around any MathML element, but even
then, table columns would still sometimes need to act as alignment
scopes, and since they are not elements themselves, but rather are
made from corresponding parts of the content of several
mtr elements, they could not individually be the
content of an alignment scope element.

An mtable element can be given the attribute
alignmentscope="false" to cause
its columns not to act as alignment scopes. This is discussed further at
the end of this section. Otherwise, the discussion in this section assumes
that this attribute has its default value of "true".

3.5.5.2 Specifying alignment groups

To cause alignment, it is necessary to specify, within each
expression to be aligned, the points to be aligned with corresponding
points in other expressions, and the beginning of each alignment
group of sub-expressions that can be horizontally shifted as a
unit to effect the alignment. Each alignment group must contain one
alignment point. It is also necessary to specify which expressions in
the column have no alignment groups at all, but are affected only by
the ordinary column alignment for that column of the table, i.e. by
the columnalign attribute, described elsewhere.

The alignment groups start at the locations of invisible
maligngroup elements, which are rendered with
zero width when they occur outside of an alignment scope, but within
an alignment scope are rendered with just enough horizontal space to
cause the desired alignment of the alignment group that follows
them. A simple algorithm by which a MathML application can achieve this is given
later. In the example above, each equation would have one
maligngroup element before each coefficient,
variable, and operator on the left-hand side, one before the
= sign, and one before the constant on the right-hand
side.

In general, a table cell containing nmaligngroup elements contains n
alignment groups, with the ith group consisting of the
elements entirely after the ith
maligngroup element and before the
(i+1)-th; no element within the table cell's content
should occur entirely before its first
maligngroup element.

Note that the division into alignment groups does not
necessarily fit the nested expression structure of the MathML
expression containing the groups - that is, it is permissible for one
alignment group to consist of the end of one
mrow, all of another one, and the beginning of a
third one, for example. This can be seen in the MathML markup for the
present example, given at the end of this section.

The nested expression structure formed by mrows
and other layout schemata should reflect the mathematical structure of the
expression, not the alignment-group structure, to make possible optimal
renderings and better automatic interpretations; see the discussion of
proper grouping in section Section 3.3.1 Horizontally Group Sub-Expressions
(mrow). Insertion of
alignment elements (or other space-like elements) should not alter the
correspondence between the structure of a MathML expression and the
structure of the mathematical expression it represents.

Although alignment groups need not
coincide with the nested expression structure of layout schemata,
there are nonetheless restrictions on where an maligngroup
element is allowed within a table cell. The maligngroup
element may only be contained within elements (directly or indirectly) of the following types
(which are themselves contained in the table cell):

an mrow element, including an inferred
mrow such as the one formed by a multi-argument
mtd element;

an mstyle element;

an mphantom element;

an mfenced element;

an maction element, though only its
selected sub-expression is checked;

a semantics element.

These restrictions are intended to ensure that alignment can be
unambiguously specified, while avoiding complexities involving things
like overscripts, radical signs and fraction bars. They also ensure
that a simple algorithm suffices to accomplish the desired
alignment.

Note that some positions for an maligngroup
element, although legal, are not useful, such as for an
maligngroup element to be an argument of an
mfenced element. When inserting an
maligngroup element before a given element in
pre-existing MathML, it will often be necessary, and always
acceptable, to form a new mrow element to contain
just the maligngroup element and the element it
is inserted before. In general, this will be necessary except when the
maligngroup element is inserted directly into an
mrow or into an element that can form an
inferred mrow from its contents. See the warning
about the legal grouping of "space-like elements" in
Section 3.2.7 Space (mspace).

For the table cells that are divided into alignment groups, every
element in their content must be part of exactly one alignment group,
except the elements from the above list that contain
maligngroup elements inside them, and the
maligngroup elements themselves. This means
that, within any table cell containing alignment groups, the first
complete element must be an maligngroup element,
though this may be preceded by the start tags of other elements.

This requirement removes a potential confusion about how to align
elements before the first maligngroup element,
and makes it easy to identify table cells that are left out of their
column's alignment process entirely.

Note that it is not required that the table cells in a column that
are divided into alignment groups each contain the same number of
groups. If they don't, zero-width alignment groups are effectively
added on the right side of each table cell that has fewer groups than
other table cells in the same column.

3.5.5.3 Table cells that are not divided into alignment groups

Expressions in a column that are to have no alignment groups
should contain no maligngroup
elements. Expressions with no alignment groups are aligned using only
the columnalign attribute that applies to the table
column as a whole, and are not affected by the groupalign
attribute described below. If such an expression is wider than the
column width needed for the table cells containing alignment groups,
all the table cells containing alignment groups will be shifted as a
unit within the column as described by the columnalign
attribute for that column. For example, a column heading with no
internal alignment could be added to the column of two equations given
above by preceding them with another table row containing an
mtext element for the heading, and using the
default columnalign="center" for the table, to
produce:

equations with aligned variables
8.44x + 55 y = 0
3.1 x - 0.7y = -1.1

or, with a shorter heading,

some equations
8.44x + 55 y = 0
3.1 x - 0.7y = -1.1

3.5.5.4 Specifying alignment points using malignmark

Each alignment group's alignment point can either be specified by
an malignmark element anywhere within the
alignment group (except within another alignment scope wholly
contained inside it), or it is determined automatically from the
groupalign attribute. The groupalign
attribute can be specified on the group's preceding
maligngroup element or on its surrounding
mtd, mtr, or
mtable elements. In typical cases, using the
groupalign attribute is sufficient to describe the
desired alignment points, so no malignmark
elements need to be provided.

The malignmark element indicates that the
alignment point should occur on the right edge of the preceding
element, or the left edge of the following element or character,
depending on the edge attribute of
malignmark. Note that it may be necessary to
introduce an mrow to group an
malignmark element with a neighboring element,
in order not to alter the argument count of the containing
element. (See the warning about the legal grouping of "space-like
elements" in Section 3.2.7 Space (mspace)).

When an malignmark element is provided within an
alignment group, it can occur in an arbitrarily deeply nested element
within the group, as long as it is not within a nested alignment scope. It
is not subject to the same restrictions on location as maligngroup elements. However, its immediate
surroundings need to be such that the element to its immediate right or
left (depending on its edge attribute) can be
unambiguously identified. If no such element is present, renderers should
behave as if a zero-width element had been inserted there.

For the purposes of alignment, an element X is considered to be to the
immediate left of an element Y, and Y to the immediate right of X, whenever
X and Y are successive arguments of one (possibly inferred) mrow element, with X coming before Y. In the case of
mfenced elements, MathML applications should evaluate this
relation as if the mfenced element had been
replaced by the equivalent expanded form involving mrow. Similarly, an maction
element should be treated as if it were replaced by its currently selected
sub-expression. In all other cases, no relation of "to the immediate
left or right" is defined for two elements X and Y. However, in the
case of content elements interspersed in presentation markup, MathML applications
should attempt to evaluate this relation in a sensible way. For example, if
a renderer maintains an internal presentation structure for rendering
content elements, the relation could be evaluated with respect to
that. (See Chapter 4 Content Markup and Chapter 5 Combining Presentation and Content Markup for further
details about mixing presentation and content markup.)

malignmark elements are allowed to occur within
the content of token elements, such as mn,
mi, or mtext. When this
occurs, the character immediately before or after the
malignmark element will carry the alignment
point; in all other cases, the element to its immediate left or right
will carry the alignment point. The rationale for this is that it is
sometimes desirable to align on the edges of specific characters
within multi-character token elements.

If there is more than one malignmark element
in an alignment group, all but the first one will be ignored. MathML
applications may wish to provide a mode in which they will warn about
this situation, but it is not an error, and should trigger no warnings
by default. The rationale for this is that it would
be inconvenient to have to remove all
unnecessary malignmark elements from
automatically generated data, in certain cases, such as when they are
used to specify alignment on "decimal points" other than the '.'
character.

malignmark has one attribute,
edge, which specifies whether the alignment point will be
found on the left or right edge of some element or character. The
precise location meant by "left edge" or "right edge" is discussed
below. If edge="right", the alignment point is the right
edge of the element or character to the immediate left of the
malignmark element. If edge="left",
the alignment point is the left edge of the element or character to
the immediate right of the malignmark
element. Note that the attribute refers to the choice of edge rather
than to the direction in which to look for the element whose edge will
be used.

For malignmark elements that occur within
the content of MathML token elements, the preceding or following
character in the token element's content is used; if there is no such
character, a zero-width character is effectively inserted for the
purpose of carrying the alignment point on its edge. For all other
malignmark elements, the preceding or following
element is used; if there is no such element, a zero-width element is
effectively inserted to carry the alignment point.

The precise definition of the "left edge" or "right edge" of a
character or glyph (e.g. whether it should coincide with an edge of
the character's bounding box) is not specified by MathML, but is at
the discretion of the renderer; the renderer is allowed to let the
edge position depend on the character's context as well as on the
character itself.

For proper alignment of columns of numbers (using groupalign values of "left", "right", or "decimalpoint"), it is
likely to be desirable for the effective width (i.e. the distance between
the left and right edges) of decimal digits to be constant, even if their
bounding box widths are not constant (e.g. if "1" is narrower
than other digits). For other characters, such as letters and operators, it
may be desirable for the aligned edges to coincide with the bounding
box.

The "left edge" of a MathML element or alignment group
refers to the left edge of the leftmost glyph drawn to render the element
or group, except that explicit space represented by mspace or mtext elements
should also count as "glyphs" in this context, as should
glyphs that would be drawn if not for mphantom
elements around them. The "right edge" of an element or
alignment group is defined similarly.

maligngroup has one attribute,
groupalign, which is used to determine the position of
its group's alignment point when no malignmark
element is present. The following discussion assumes that no
malignmark element is found within a group.

In the example given at the beginning of this section, there is one
column of 2 table cells, with 7 alignment groups in each table cell;
thus there are 7 columns of alignment groups, with 2 groups, one above
the other, in each column. These columns of alignment groups should be
given the 7 groupalign values "decimalpoint left left
decimalpoint left left decimalpoint", in that order. How to specify
this list of values for a table cell or table column as a whole, using
attributes on elements surrounding the
maligngroup element is described later.

If groupalign is "left",
"right", or "center", the alignment point is
defined to be at the group's left edge, at its right edge, or halfway
between these edges, respectively. The meanings of "left edge"
and "right edge" are as discussed above in relation to malignmark.

If groupalign is "decimalpoint",
the alignment point is the right edge of the last character before the
decimal point. The decimal point is the first "." character
(ASCII 0x2e) in the first mn element found along
the alignment group's baseline. More precisely, the alignment group is
scanned recursively, depth-first, for the first mn
element, descending into all arguments of each element of the types
mrow (including inferred
mrows), mstyle,
mpadded, mphantom,
mfenced, or msqrt,
descending into only the first argument of each "scripting" element
(msub, msup,
msubsup, munder,
mover, munderover,
mmultiscripts) or of each
mroot or semantics element,
descending into only the selected sub-expression of each
maction element, and skipping the content of all
other elements. The first mn so found always
contains the alignment point, which is the right edge of the last
character before the first decimal point in the content of the
mn element. If there is no decimal point in the
mn element, the alignment point is the right edge
of the last character in the content. If the decimal point is the
first character of the mn element's content, the
right edge of a zero-width character inserted before the decimal point
is used. If no mn element is found, the right
edge of the entire alignment group is used (as for
groupalign="right").

In order to permit alignment on decimal points in
cn elements, a MathML application can convert a
content expression into a presentation expression that renders the
same way before searching for decimal points as described above.

If characters other than "." should be used as
"decimal points" for alignment, they should be preceded by malignmark elements within the mn token's content itself.

For any of the groupalign values, if an explicit
malignmark element is present anywhere within
the group, the position it specifies (described earlier) overrides the
automatic determination of alignment point from the
groupalign value.

3.5.5.7 Inheritance of groupalign values

It is not usually necessary to put a groupalign
attribute on every maligngroup element. Since
this attribute is usually the same for every group in a column of
alignment groups to be aligned, it can be inherited from an attribute
on the mtable that was used to set up the
alignment scope as a whole, or from the mtr or
mtd elements surrounding the alignment group. It
is inherited via an "inheritance path" that proceeds from
mtable through successively contained
mtr, mtd, and
maligngroup elements. There is exactly one
element of each of these kinds in this path from an
mtable to any alignment group inside it. In
general, tThe value of groupalign will be
inherited by any given alignment group from the innermost element
that surrounds the alignment group and provides an explicit
setting for this attribute. For example, if an
mtable element specifies values for groupalign and
a maligngroup element within the table also specifies an
explicit groupalign value, then then the value from the
maligngroup takes priority.

Note, however, that each mtd element needs, in
general, a list of groupalign values, one for each
maligngroup element inside it, rather than just
a single value. Furthermore, an mtr or
mtable element needs, in general, a list of lists
of groupalign values, since it spans multiple
mtable columns, each potentially acting as an
alignment scope. Such lists of group-alignment values are specified
using the following syntax rules:

As described in Section 2.4.4 MathML Attribute Values, | separates
alternatives; + represents optional repetition (i.e. 1 or
more copies of what precedes it), with extra values ignored and the
last value repeated if necessary to cover additional table columns or
alignment group columns; '{' and '}'
represent literal braces; and ( and ) are
used for grouping, but do not literally appear in the attribute
value.

The permissible values of the groupalign attribute of the
elements that have this attribute are specified using the above
syntax definitions as follows:

Element type

groupalign attribute syntax

default value

mtable

group-alignment-list-list

{left}

mtr

group-alignment-list-list

inherited from mtable attribute

mtd

group-alignment-list

inherited from within mtr attribute

maligngroup

group-alignment

inherited from within mtd attribute

In the example near the beginning of this section, the group
alignment values could be specified on every mtd
element using groupalign = "decimalpoint left left
decimalpoint left left decimalpoint", or on every
mtr element using groupalign =
"{decimalpoint left left decimalpoint left left decimalpoint}", or
(most conveniently) on the mtable as a whole
using groupalign = "{decimalpoint left left decimalpoint
left left decimalpoint}", which provides a single braced list of
group-alignment values for the single column of expressions to be
aligned.

3.5.5.8 MathML representation of an alignment example

The above rules are sufficient to explain the MathML representation
of the example given near the start of this section.
To repeat the example, the desired rendering is:

3.5.5.9 Further details of alignment elements

The alignment elements maligngroup and
malignmark can occur outside of alignment
scopes, where they are ignored. The rationale behind this is that in
situations in which MathML is generated, or copied from another
document, without knowing whether it will be placed inside an
alignment scope, it would be inconvenient for this to be an error.

An mtable element can be given the attribute alignmentscope="false" to cause its
columns not to act as alignment scopes. In general, this attribute has the
syntax (true | false) +; if its value is a list of boolean
values, each boolean value applies to one column, with the last value
repeated if necessary to cover additional columns, or with extra values
ignored. Columns that are not alignment scopes are part of the alignment
scope surrounding the mtable element, if there is
one. Use of alignmentscope="false" allows nested tables to contain malignmark elements for aligning the inner table in the
surrounding alignment scope.

As discussed above, processing of alignment for content elements is
not well-defined, since MathML does not specify how content elements
should be rendered. However, many MathML applications are likely to find it
convenient to internally convert content elements to presentation
elements that render the same way. Thus, as a general rule, even if a
renderer does not perform such conversions internally, it is
recommended that the alignment elements should be processed as if it
did perform them.

A particularly important case for renderers to handle gracefully is the
interaction of alignment elements with the matrix
content element, since this element may or may not be internally converted
to an expression containing an mtable element for
rendering. To partially resolve this ambiguity, it is suggested, but not
required, that if the matrix element is converted
to an expression involving an mtable element, that
the mtable element be given the attribute alignmentscope="false", which will
make the interaction of the matrix element with the
alignment elements no different than that of a generic presentation element
(in particular, it will allow it to contain malignmark elements that operate within the alignment
scopes created by the columns of an mtable that
contains the matrix element in one of its table
cells).

The effect of alignment elements within table cells that have
non-default values of the columnspan or rowspan attributes is not specified, except that such
use of alignment elements is not an error. Future versions of MathML may
specify the behavior of alignment elements in such table cells.

The effect of possible linebreaking of an mtable
element on the alignment elements is not specified.

3.5.5.10 A simple alignment algorithm

A simple algorithm by which a MathML application can perform the
alignment specified in this section is given here. Since the alignment
specification is deterministic (except for the definition of the left
and right edges of a character), any correct MathML alignment
algorithm will have the same behavior as this one. Each
mtable column (alignment scope) can be treated
independently; the algorithm given here applies to one
mtable column, and takes into account the
alignment elements, the groupalign attribute described in
this section, and the columnalign attribute described
under mtable (Section 3.5.1 Table or Matrix
(mtable)).

First, a rendering is computed for the contents of each table cell
in the column, using zero width for all
maligngroup and malignmark
elements. The final rendering will be identical except for horizontal
shifts applied to each alignment group and/or table cell. The
positions of alignment points specified by any
malignmark elements are noted, and the remaining
alignment points are determined using groupalign
values.

For each alignment group, the horizontal positions of the left
edge, alignment point, and right edge are noted, allowing the width of
the group on each side of the alignment point (left and right) to be
determined. The sum of these two "side-widths", i.e. the sum of the
widths to the left and right of the alignment point, will equal the
width of the alignment group.

Second, each column of alignment groups, from left to right, is
scanned. The ith scan covers the ith
alignment group in each table cell containing any alignment
groups. Table cells with no alignment groups, or with fewer than
i alignment groups, are ignored. Each scan computes two
maximums over the alignment groups scanned: the maximum width to the
left of the alignment point, and the maximum width to the right of the
alignment point, of any alignment group scanned.

The sum of all the maximum widths computed (two for each column of
alignment groups) gives one total width, which will be the width of
each table cell containing alignment groups. Call the maximum number
of alignment groups in one cell n; each such cell's width
is divided into 2n adjacent sections, called
L(i) and R(i) for i from 1 to
n, using the 2n maximum side-widths computed
above; for each i, the width of all sections called
L(i) is the maximum width of any cell's ith
alignment group to the left of its alignment point, and the width of
all sections called R(i) is the maximum width of any
cell's ith alignment group to the right of its alignment
point.

The alignment groups are then positioned in the unique way that
places the part of each ith group to the left of its
alignment point in a section called L(i), and places the
part of each ith group to the right of its alignment
point in a section called R(i). This results in the
alignment point of each ith group being on the boundary
between adjacent sections L(i) and R(i), so
that all alignment points of ith groups have the same
horizontal position.

The widths of the table cells that contain no alignment groups
were computed as part of the initial rendering, and may be different
for each cell, and different from the single width used for cells
containing alignment groups. The maximum of all the cell widths (for
both kinds of cells) gives the width of the table column as a
whole.

The position of each cell in the column is determined by the
applicable part of the value of the columnalign attribute
of the innermost surrounding mtable,
mtr, or mtd element that
has an explicit value for it, as described in the sections on those
elements. This may mean that the cells containing alignment groups
will be shifted within their column, in addition to their alignment
groups having been shifted within the cells as described above, but
since each such cell has the same width, it will be shifted the same
amount within the column, thus maintaining the vertical alignment of
the alignment points of the corresponding alignment groups in each
cell.

3.6 Enlivening Expressions

3.6.1 Bind Action to Sub-Expression (maction)

There are many ways in which it might be desirable to make
mathematical content active. Adding a link to a MathML sub-expression
is one basic kind of interactivity. See Section 7.1.4 Mixing and Linking MathML and HTML.
However, many other kinds of interactivity cannot be easily
accommodated by generic linking mechanisms. For example, in lengthy
mathematical expressions, the ability to "fold"
expressions might be provided, i.e. a renderer might allow a reader to
toggle between an ellipsis and a much longer expression that it
represents.

To provide a mechanism for binding actions to expressions, MathML
provides the maction element. This element accepts any
number of sub-expressions as arguments.

By default, MathML applications that do not recognize the specified
actiontype should render the selected sub-expression as
defined below. If no selected sub-expression exists, it is a MathML
error; the appropriate rendering in that case is as described in
Section 7.2.2 Handling of Errors.

Since a MathML application is not
required to recognize any particular actiontypes, an
application can be in MathML conformance
just by implementing the above-described default behavior.

The selection attribute is provided for those
actiontypes that permit someone viewing a document to select one of
several sub-expressions for viewing. Its value should be a positive
integer that indicates one of the sub-expressions of the
maction element, numbered from 1 to the number of
children of the element. When this is the case, the sub-expression so
indicated is defined to be the "selected sub-expression" of the
maction element; otherwise the "selected
sub-expression" does not exist, which is an error. When the
selection attribute is not specified (including for
actiontypes for which it makes no sense), its default value is 1, so
the selected sub-expression will be the first sub-expression.

Furthermore, as described in Chapter 7 The MathML Interface, if a MathML
application responds to a user command to copy a MathML sub-expression to
the environment's "clipboard", any maction elements present in what is copied should
be given selection attributes that correspond to their selection
state in the MathML rendering at the time of the copy command.

A suggested list of actiontypes and their associated
actions is given below. Keep in mind, however, that this list is
mainly for illustration, and recognized values and behaviors will vary
from application to application.

For this action type, a renderer would alternately display the
given expressions, cycling through them when a reader clicked on the
active expression, starting with the selected expression and updating
the selection attribute value as described above.
Typical uses would be for exercises in education, ellipses in long
computer algebra output, or to illustrate alternate notations. Note
that the expressions may be of significantly different size, so that
size negotiation with the browser may be desirable. If size
negotiation is not available, scrolling, elision, panning, or some
other method may be necessary to allow full viewing.

<maction actiontype="statusline"> (expression) (message) </maction>

In this case, the renderer would display the expression in
context on the screen. When a reader clicked on the expression or
moved the mouse over it, the renderer would send a rendering of the
message to the browser statusline. Since most browsers in the
foreseeable future are likely to be limited to displaying text on their
statusline, authors would presumably use plain text in an
mtext element for the message in most circumstances.
For non-mtext messages, renderers might provide a
natural language translation of the markup, but this is not
required.

<maction actiontype="tooltip"> (expression) (message) </maction>

Here the renderer would also display the expression in context
on the screen. When the mouse pauses over the expression for a long
enough delay time, the renderer displays a rendering of the message in
a pop-up "tooltip" box near the expression. These message boxes are
also sometimes called "balloon help" boxes. Presumably authors would
use plain text in an mtext element for the message
in most circumstances. For non-mtext messages,
renderers may provide a natural language translation of the markup if
full MathML rendering is not practical, but this is not
required.

In this case, a renderer might highlight the enclosed expression on
a "mouse-over" event. In the example given above,
non-standard attributes from another namespace are being used to pass
additional information to renderers that support them, without violating the MathML DTD (see
Section 7.2.3 Attributes for unspecified data). The my:color attribute
changes the color of the characters in the presentation, while the
my:background attribute changes the color of the background
behind the characters.